I've right clicked on the castle and church in each province and made my own baronies.. wait.. vassals? Whatever it is. But I'm still "the guy in charge of the province" when I click on it.
These are "baron" tier titles. Barons usually hold a single town or castle that is smaller than the CK2 province size. So a province in CK2 consists of several baronies. The "count" owns the main holding of the province and is the "boss", so the barons are his de-jure vasalls.
If you assign the "count" title of the province to any of the province barons, their holding will become the main holding (it will switch to the big slot).
It is good to assign barons to other holdings in a province, because each holding counts against your demesne limit and it is usually better to get several count titles to increase the number of provinces you own. In addition, barons are only simulated rudimentarily, so they will not plot or revolt, making them fantastic vassals.
Lastely, you will receive a massive penalty for holding a church or town yourself, so it is usually best to just assign these to a random characer (via the RMB menu) or an unlanded character (which will then be turned into the optimal character type for the holding, e.g. a bishop if it is a church).
The only time you might want to own multiple holdings in a province is when it has 5 or 6 slots and you can fill them with castles. Why? Because you can assign your council members to boost levy size & regenration (marshal) or tax income (treasurer) in a single province. If you have a brilliant marshal, his boost will apply to ALL of these castles, and receiving a +50% levy size bonus and +50% tax on 3-4 fully expanded castles (you can't have more since one of each kind - castle, church, town must exist before you can start to build copies) can be more powerful than having 3 additional provinces with barons.
Can I put somebody in charge there who likes me? One of my dudes? How? How can I see what dudes I have available?
There is a button in your character portrait menu called "Court". This shows all available unlanded characters at your court. "Vassals" will show all landed characters in your realm with a title. You can also check out relatives via the "Family" button. You can also sort the list by relation value to pick the most loyal guys.
To assign a title to a character YOU must own it. If it belongs to another character, you must first conquer (outside your realm) or revoke (inside your realm) it. Select the portrait of the desired character, then click on the diplomacy scroll below the Coat of Arms and select "Grant Landed Title". You will then receive a list of eligible titles to give to the person. Note: If you want to give away a duke title or higher, you must first give him a count title, so that he receives a province.
I was also told not to put any of my children, especially the heir in charge of anything.. because then when I die things could fall apart? What's that all about?
If you give a title to your young son, he will leave your court to govern the holding(s). This means you can no longer tutor him yourself and you can no longer arrange marriages. He will also end up with a regent - and if there is any chance that the regent might inherit the title, he is more than ready to murder your son.
And what's the ideal way of raising armies? I declare war, then right click on random provinces and raise levies, then merge the armies into larger units, which I then move into the combat zone. I saw that you can do the same thing from the "warfare" icon up top, is that interface better to use?
Yes, it is. A thousand times. You can use it to raise all your own levies (top button), to raise all levies from your provinces (including your barons, middle left button) or all levies from vassals within your realm (only if you are a duke or higher, this includes your vassal counts that have their own provinces, middle row, middle button).
By holding CTRL you will only raise these levies in provinces where no enemies are present (if you don't do that they will instantly end up in combat with minimal morale and usually die).
You can also use the buttons on the right to dismiss certain levies (e.g. all levies from your counts to just keep your own troops). This is useful because your vassals will receive a stacking opinion penalty if you use their troops for too long.
I try not to hire mercenaries, because I did a couple times and when I ran out of money they turned on me.. So I had to reload and try my hand at the invasion again. (Isn't trial and error fun?)
Mercenaries are extremely powerful, because you get a lot of combat power for your buck. They are ideal if you want/have to win a deceisive battle - just hire them, wait for their morale to tick up and then charge into the fray, win and disband once your own troops can handle the enemy. They are also great if you want to save your own manpower for sieges/assaults.
Lastely, unlike your normal levies they will reinforce over time, so IF you can pay them, they are great for use in far away regions (e.g. oversea or during a crusade).Their monthly cost scales with their number.
What should I be paying attention to in terms of my economy? How can I improve it? Should I be building and/or upgrading buildings in my provinces?
Build towns in all your holdings asap. Try to grab more count titles to increase the number of provinces you own. Increase town tax levels to get more money from your cities/mayors. REDUCE church taxes and levies to make bishops happy, because they will give all their money to the pope if they like him more than you.
What else should I be doing? I realize that this game is a bit open ended so you can do whatever you want, but I bet I'm missing out on some stuff that I should be doing but just don't know about yet.
Just immerse yourself and enjoy the story that unfolds. Usual goal is to climb up the ladder and become a king or even emperor. But you can also set other goals, like installing members of your dynasty in Europe's kingdoms.
And as for my son, he's 7. Once he's 15 or whatever, should I be looking to marry him off to an Austrian or Hungarian princess or something? Would that give me a PU?
No. There are no PUs in this game. Characters can receive "claims" on other titles by marrying the right persons, but you will usually have to enforce them via war (think of them as a Casus Belli).
Btw, I highly recommend to start as an Irish count in the 1066 scenario if you want to learn the ropes. I'd recommend the count of Leinster - your son and heir is count of Dublin, so once you die he will inherit your county and you have two provinces under your control. Then try to become a duke and take the connected de-jure titles. Once you are strong enough, fabricate claims on the northern counties and bring enough provinces under your control (directly or as vassals) to become king of Ireland.
After that, you can try to expand into Scotland to ultimately become king of Scotland (and later create the empire of Britannia by conquering enough of England). Or you can go on crusades to fight against the infidels.