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.