How can you accumulate early wealth? If I expand my empire, my money goes down the drain and I'm stuck with a research slider at 50 to 40%, making my civilization lag behind everyone else's. Or If I don't expand I'm left with a few cities clumped together and that's no good either. Is there any way to keep expanding without research being painfully slow.

basically axe = expansion

and if you still not satisfied then:
lots of cottages = money + research
specialist = research

50% is the ideal research rate. Any higher means you're not expanding fast enough.

Your problem is probably not enough workers and cottages.

Really? I always thought I was doing horrible.

The 50% assumes you are in a state of expansion and either war preparation or engagement I think. It is definitely worthwhile to play that way...many people get too focused on building buildings and such, and while that's fun, it isn't the most cutthroat way to annihilate your enemies and expand your civ.

Really? I've never consistently run at 50%...

Either I'm raking in the moolah so running at 70%+, or I've been knocking out neighbouring civs for a few centuries and my tech rate falls through the floor (losing money at 0% and raising funds purely from razing cities...good stuff ;) )

Every game is different. I usually play Monarch/Normal/Space Race or Domination games, so if that's anything like what your games are my advice may be helpful.

Usually, in the early game a low slider is almost unavoidable if you are trying to expand your empire. If you get to the modern age, you should be able to run at around 70-80 and turn a nice profit. You make up for that slow early progress in a number of ways:

1) the obvious way- accumulate cash from taking over cities, selling techs after currency. Use the cash to temporarily keep researching at a good level while your cottages develop. This isn't necessary, but often expanding to become a large empire requires some early conquests.

2) SPECIALISTS. Switch to caste system if you like, or build libraries and run scientists. Early scientists make a huge difference in research. You can run the slider at 0 if you have enough scientists. Bulbing is also a good way to catch up, particularly if you are philosophical and can churn them out quickly. In the late game, golden ages can be a huge boom for you as well.

3) Cottages. They start small but the idea is early expansion + mid game switch to 100% cottage economy = massive late game economy. My best games, I'm running at a deficit early on but if you have 25 cities with a bunch of towns, in the late game when you're trying to build your space ship, no tech should take more than about 4 or 5 turns, and you'll easily catch up (and overtake) the AI. This is necessary as the cheating bastard AI builds spaceship parts much quicker than you on the higher levels.

So basically, don't worry about that low slider, as long as it's not low the entire game.

Early wealth comes from founding a religion, building its Shrine, and spreading said religion. 1g/turn/city is great.

I'm not one to run a low research rate... I find that if my research rate drops below 80%, I'm soon far behind techwise. A good way to keep up cash flow is to keep capturing AI cities, I find. :)

50% is the ideal research rate. Any higher means you're not expanding fast enough.

Your problem is probably not enough workers and cottages.

agreed
In other words, 50% of a large commerce basis is great.
Of course 50% of the 8 commerce from the palace isn't so great.

There is actually very good way to make money early and a lot of it. Wonders!
If you fail to build wonder first, all your hammers goes back as money.

This tip really works only if you have bunus to producing wonders. This includes Industrious trait and/or stone, marble, copper.

How this works: Start building for example Stonehenge if you have stone hooked by. That will give you 100% bonus on hammers. If you fail to build stonehenge you will get eventually 2gold for every hammer invested in wonder. I sometimes start to build wonders just for that reason.

Now, if you are industrious and you have stone hooked that's 2.5gold for every hammer. 1 grassland hill will give you 7.5 gold per turn as long as your building wonder with no intention to complete it.

I usually do this if I'm waiting for city to grow after whipping.

Personally, I'm not too big on gathering too much early wealth. All it seems to do is attract the attention of someone who wants it from you. Someone will be inspired to demand tribute. If you don't have it, they don't bother. I invest it in expansion and my military. Should someone still want some, I'm in a position to pay them with bladed iron instead of coined gold. :D

On prince level, marathon speed, huge random maps, wrlords. If I have a small empire I try to shoot for a profit at 70% before I build another city. If I am expanding militarily I do not worry about the science slider because of the plunder I get. After absorbing several mature AI cities the science slider can go to 40-50% but still producing similar research as the smaller 70% empire. The idea at that point is to start building (courthouses/market) until you get the science slider to what it was before the expansion. Then expamd again.

If you're running a Specialist Economy, war provides another huge boost to your coffers–razing cottages. You want farms on those squares anyway, and well-worked cottages are big cashoola.

You can't just base everything on your slider. Slider affects all your cities, so just because it's on 10% in one case does not mean you are making less than when you have it on 90% in another.

Around 90% of my turns in my games I keep the slider on 0% anyhow, and still manage ok. Sometimes I have to turn the spending tap back on when something goes wrong, e.g. universal sufferage civic is forced by UN or some such thing..