How can stealing be done for fun and profit?

There are a number of parts to this. First of all, the pure mechanics of looting stuff.

Stealing things would often come under the Burglary and Stealth skills, but other skills might get involved too.

Find a likely target (Contacts), observe it to find the best time and way to break in (Burglary, creating an aspect to use later if needed), pick the lock to get in (Burglary), avoid being spotted by the security patrols (Stealth), open the safe (Burglary) and then get away with the proceeds.

Fencing the valuables needs someone to buy the item (a Contacts roll is likely to solve that one) and then to convince them that they should give the character a good deal (and not cheat them, betray them, or decide that they are too dangerous to go anywhere near).

So Resources hasn't actually come into it until now. You use Resources to buy things, not to steal or sell them.

Having got the goods and found someone who will take them: What now?

The player can't simply increase Resources. It is a skill, Fate doesn't usually permanently modify skills just for plot. Skills change at Milestones, and you can use "I've pulled off several burglaries" as in-fiction justification for the changes in skills you select at Milestones.

You can, however, give them an aspect. "Windfall from the Camberley Caper" can sit around until the character needs some cash and then be invoked for a bonus to a resources roll.

But mechanics aside, the real thing you need to do is to get you and the player on the same page about what sort of game you are doing.

"wanting to steal everything not nailed down and sell it off for cash" does sound like they are trying to play D&D. Now, D&D is a game where money and experience are closely linked. Get money, buy magic, kill monster, loot horde, buy bigger magic, kill bigger monster, and so on. That isn't Fate.

D&D is, at its core, a game about killing things and taking their stuff. That is what the mechanics support and that is what roleplay tends to be built on top of in that game.

Fate is a game that is much more focused on telling stories. There are plenty of stories about thieves, but the focus isn't on "wanting to steal everything not nailed down and sell it off for cash". The stories focus on the big jobs and leave the small stuff off screen.

I think you've got it a bit wrong - stealing stuff is totally under burglary.

If your player wanted to increase his wealth through theft, he could continue to steal things and occasionally play a scene with fencing it that would involve a social roll, like deceive or rapport.

Then, at a milestone, he could bump his resources by trading it with a higher skill to reflect the effort he's put into increasing his available resources.

Just remember that one theft isn't going to do it. Stealing a very valuable item is as much trouble as it is help! And even if you don't recognize the value, you might steal something that means a great deal to the real owner. Don't hesitate to give your player trouble that he asks for so plainly!

But changing your circumstances is what milestones are for - they let you reflect the story in your character.

In a game I've mastered, my players managed to acquire some cash, but that was one-time event. So it became an aspect "a nice purse of gold" with some free invocations that could be used in Resource rolls (or actually Burglary because of Black Market stunt).

In a similar way you can handle it mechanically as a "Create advantage" with Steal which can be used for future Resource rolls.