I don't think it's a good idea to stick with a civilization which you feel is worse than it should be. However, one very powerful learning technique - is repetition. There is a saying in my country: "repetition - is mother of learning". Civilization 5 is not an exception - it was already mentioned (probably repeatedly) in this forum that starting your game over and over again from turn 1 (same map - using a turn-1 save, that is), while aiming to get better results than you did before, - is an excellent learning method for Civ 5. Perhaps, try it?
And then, there are two things which i think you need (and probably are going to get mostly on its own in the process of playing the game). I'll list 2nd thing 1st, because the 1st thing requires somewhat lengthy description - so it goes 2nd. Hope i am not too... crazy with that last sentence. %)
So. Second thing, - find your favorite playstyle. You'd need to try different things to realize what you like the most. Is it indeed domination - or perhaps you'd enjoy building massive gold-generating empire more? You can't be sure unless you'd try. Or, domination itself - it can still be very different playstyles "inside" domination banner. Do you go tall and raze 'em AIs, just leaving their capital cities be (which i enjoy the most whenever i go warmongering in my games, tbh) - or do you build _efficient_ wide empire, puppeting or even annexing captured cities? Do you go through serious of relatively short wars (like Persian civ would probably do) - or do you refuse peace offers until there is noone to offer you peace anymore?
Next, do you prefer to fight inside your own territory (or close to it) and "grind down" enemy forces until you can do a decisive strike upon one (or more) of their cities - or do you prefer to rush-build lots of units and do a "zerg rush"? Etc. But, you can't find your favorite playstyle if you don't know what possible efficient playstyles are - and this knowledge comes from detailed knowledge of what's possible in terms of your civilization's "base" (which is its economic and productive power). Said "base", in turn, is dependant on many other major mechanics of the game, - and any good player must be at least somewhat familiar with each. Only then you'll be able to find out what your favorite playstyle actually is. Which brings us to the
First thing: learn how to use _all_ major game mechanics at your disposal, and get a habit to keep managing every major mechanic substantially well. You see, every major concept there is - can (and should!) be helping you to achieve victory. Examples:
- Trade routes. Do you always have max number of them busy and going? Do you trade with best possible trade partners (considering both profitability but also safety of trade routes)? Get a habit to maintain your trade relations in order, all the time. Good trading is a massive help;
- Growth. Do you manage your growth properly? Two standard approaches are "tall" - which is few cities which you grow as fast and as large as humanly possible; and "wide" - which is when you get many cities (~dozen or more), but intentionally limit their growth and at some point, stall it. So you see, if you have few cities but do not manage to grow them VERY fast - you'll lose; if you have many cities but do not manage to limit/stall their growth appropriatly - you'll also lose (happiness issues aplenty). Furthermore, managing solid growth of wide empite involves certain social policies, right ideaology and tenets - it's possible to reduce additional unhappiness per every extra city in a wide empite to truly miniscule amoutns (like ~ +1 extra unhappiness for every extra city of ~10 citizens);
- Religion. Another big helper if/when done right. Find beliefs which suit your plan the most (wide empires need much different ones than tall empires). Make a habit to find pantheon early (normally through intensive scouting early in the game, for which reasons build at _very_ least 1 scout extremely early) - ruins and possibly religions CSs should give enough faith. Once you get a religion, make it a habit to spread it with at least a few missionaries before spending your faith to anything else - the sooner you will have _many_ cities (yours and AI's) following your religion, the bigger boost you'll be having from your religion for the rest of the game;
- City-states. Very massive help when managed well. Do their quests as much as possible, keep an eye on every message from city-states every turn - don't skip paying attention to any of them. Spread your religion to them - this cuts the rate at which your influence degrades, substantially. If you have some military CSs as friends/allies - never disable their unit-giving, and gift any unneeded units you got from them - to a city-state you need the influence with the most;
- Science. Always keep in mind at least 2 "next" tiers of technologies - in detail. Always define which particular tech among next 1...3 tiers you want to get ASAP, and work towards it. Once done, define next target tech in the same manner and go for it, rinse repeat. Most of the time, any technologies which give you any large boost to scientific power - are a big priority; getting libraries, univercities, observatories and public schools as early as humanly possible - is a most powerful research approach for vast majority of games;
- Culture. While it sounds somewhat secondary to many players, in fact it is also of key importance. Make a habit to have +2 culture building in all your cities as fast as humanly possible. Build more +culture buildings when you can. Always attempt to get big culture-boosting wonders and policies (large +% in all cities is the best) - these are quite important. Reason for this - is your territory growth, which gets faster the more +culture you have. Larger territory means more tiles which your cities can work (even if no AIs are nearby - some city state or two is most likely is), but more than that, your territory can grow up to 5 tiles around all your cities enetually - and this means more space where your troops can heal at a dramatically increased rate, more permanently visible area, longer time it takes for any AI surprise attack to actually reach your cities, more safety for your trade routes, more chances to get mid-game and late-game resources (coal, oil, alluminium, uranium), all of which are quite important, etc;
- Production. In its own way, production is the _most_ important thing there is. Because vast majority of everything you create - comes from it (yes, it's possible to buy quite alot if one has massively powerful economy, however, it's still a little in compare to how much one creates via production). Buildings, units, wonders, things like international games and ISS - the more production you got, the more of everything you'll have (or to put it the other way - the _faster_ you'll get everything you need/want). So always remember to look for things which increase your production especially much. Personally, i nearly always take +1% per-citizen (max 15%) from religion, for example. Perhaps most important thing about production though - is to always remember about perhaps _the_ most powerful production booster, which is city growth. Even for wide empires, where growth is totally stalled for most cities at some point - until this point is reached, faster city growth is the most powerful thing to end up having a city which built/produced as much as possible.
- Diplomacy and diplomatic trade relations. This is a HUGE major mechanic in the game oh so many players completely miss. It can boost development of your civilization incredibly much. If you want to become a better player, you'll need to master this, and make good habits about it: like, never allow to have any substantial unused resources just sitting on your resource bar, doing nothing - trade them away for luxury resources you don't have (if possible), or for gold-per-turn (or straight gold when possible), or for open borders (1 luxury resource which you have multiples of - is often enough to have an AI to give you open borders without you giving it to him); also, sometimes, when an AI is guarded or hostile to another AI, few resources can actually convince him to declare war - and quite often this can be very beneficial to you, so this is also an important kind of deal possible with your extra (unused) resources. If ALL possible deals are already made at some point, but you still have more unused luxury or strategic resources - then select on AI with whom you have best chances (and most sense) to have friendly relations with, and GIFT some of your extras to him. At later point you might well be happy you have an ally. Remember all this is "temporary" - deals are 45 turns on epic speed (i play epic; 30 on normal?), so you never give away anything "forever". Then there are also research agreements, "extorting" lots of goods outta AIs through declaring war, razing a city or two of theirs, and then when they offer peace adding a city of their (usually) into the mix as a gift - replacing that city with said lots of goods; etc etc. Then there is world congress, which is a huge booster to you and a subject of its own - may be not exactly "major" game mechanic on its own, but still one which any good player needs to learn (and excellent player - to master). Diplomacy have MUCH more than it might seem it is, on the surface;
- Spying is not there until BNW, right? But if you'd ever go BNW - same thing, another major concept you should never "skip"; used right, is another big boost to your civ;
- Exploration. This one is simple. Do it. And it means, do it as early as possible, and as much as possible (without hurting other important things any much). One excellent thing - are missionaries; after initial few to set up your religion around, make more of these, and send them out as soon as you get Astronomy (if not yet) to distant lands. Found a CS there without faith? Spread religion into it and into one of nearby cities; whenever there are 2 or more not-too-far-from-each-other cities with the same religion - the thing starts to spread around on its own
- War. "War never changes", eh. War is a major game mechanic, allowing to get things otherwise impossible to get. Get a habit to have your army busy with some or other war, - if you have army at all, that is (above minimum required homeland-defense forces that is - these stay home, of course). Learn cons and pros of doing wars to neighbours - and to remote civs. And have at least a few units which are your "elite guard" - ones who do not get killed, but keep levelling up to high levels. They get time more powerful (literally - considering extra attacks, extra range, extra combat strength upgrades), and since there is a limitation of "1 unit per 1 tile" - this allows to have some "concentrated" firepower, which is great for nearly all combat situations. Guard your elite units and do not allow them to come to harm, even consider "training" them on some beaten-to-a-pulp civilization for extra eperience;
- Last but perhaps not least of my examples - is naval superiority. Here, i differ from quite many in saying that naval superiority - is much more important than air superiority. I say so, because against superior air forces, one can always bring SAMs (and later missile cruisers), which is darn effective and - relatively - cheap. But against a massive naval armada, the _only_ thing actually working - is more powerful naval armada of one's own. So, plan ahead. Do highly productive port or even more than one (depends on map). Build naval units early, and keep using them - for recon, fighting barbs, patrol, and certainly participating in any war they can take a part in. Upgrade them when possible. Get at least a pair of carriers when they get available, and use them to the fullest.
Good luck!
). Playing on a lower difficulty is also an option, but I'd rather struggle with a challenge, get better, and find myself playing up to par on a harder difficulty. Rather than bring the difficulty down and find an easy victory, I want to actually become a better player. 
