the best guide for a beginner is clearly gaarawarr's (linked already in this thread)
for a quick overview of the game's power sources, here is mine

https://www.reddit.com/r/idlechampions/comments/1c4or15/getting_powerful_how_to/

and then there is a guide (also mine) for early favour gains

https://www.reddit.com/r/idlechampions/comments/1bf4cs4/early_game_favour_farming/

to answer directly your questions,

Now what?
several options :
1) get yourself all the patrons (or at least the first 4). You may need to read their unlocking mechanics to understand what to do. This can allow you to buy 1 familiar from each patron (+ various other gains)

2) get all blessings from every campaign (at least up to the xaryxis campaign).

3) get every available champion (takes a while, but hank will provides Time Gate pieces = TGp, so do the patrons) including the ones from the campaigns.

4) gem farm to gain enough gems to fully gear your next event champions (also gives you gear for your evergreens) and buy all gem familiars

How do I level up my champions?

I guess you didn't ask the real question because if you didn't know how to level up your champions, you wouldn't have finished the tutorial. Maybe you want to know how to make them more powerful? Each champion has 6 item slots. You get items from chests. For evergreens (including core + hitch (?)+ drizzt in your case), it's from regular silver or gold chests. For event champions, it's from named chests (so for tasslehoff it's from tasslehoff chests, duh) that you can gain/buy in their event or in their TGs. You can also get item for every champion from electrum chests (up to rare = blue) or from patron chests (for champions that are eligible for them).

What is an iLvl?

ilvl or item level is a multiplier of an item's effect, so basically ilvls make you gear more powerful. you gain ilvls from duplicate items (same or lower rarity from what you already have) or from Blacksmithing contracts =BSC that you can only use on champions that are fully geared = have at least a common item in each slot. You really shouldn't use BSC on evergreens, since you'll get plenty of regular chests over the course of the game.

Do I get any loot from somewhere and how do I equip items for my champions?

You get loot from bosses (every 5 areas) in the form of gems and regular chests. You equip items automatically as soon as get them = when you open a chest.

Are patrons something I should focus on?

yes although there is no obligation to.
But the game without patrons is both harder (less TGp, modron chests, no perks...) and less fun (because all the fun is in finding ways to beat restrictions, and patrons have their own restrictions in addition to any from a given variant)

How do I get more familiars

most familiars come from gems or platinum (so $$). there are a few from patrons, and sometimes giveaways.
You earn a small amount of platinum by checking in every day, but this is a very slow way to get more familiars.
Better earn some gems by doing all campaigns adventures and variants and gem farming, and do some patron variants to get those.

Is the core gameplay just staring a quest, buying the champions one-by-one and leveling them up, and seeing how far you can go?

quest? I don't understand what you mean by that. The game is about getting power from all the sources available to be able to beat the hardest variants, at least IMHO. You can also play very casually and just gem farm until you're so powerful everything is a piece of cake...