Depends on how you want to play your game. At minimum, the unofficial patches.
For visuals, NMC textures, some lighting mods, Fallout Redesigned (for character faces) and a weather mod really perk up the game (Fellout if you want to get rid of the green tinge). I have never played this game with an ENB, so others can help you out with those. Darnified for UI.
For making the game more difficult, FWE (Fallout Wanderer's Edition) includes quite a bid of mods that all work together and is customizable. It has compatibility packages to work with the big mods like WMK (mod that adds more options to mod weapons), EVE (mod for energy weapons), etc. This mod has a lot of changes and most of the features can be turned off if you don't like them.
See You throws in some random encounters when you're sleeping.
The Quantum Pipeline makes it to where the Nuka Cola factory has a pipeline that you can fill empty bottles with for Quantum (unlimited).
DC Interiors Project creates quite a few interiors to explore within DC.
For quest mods, I've only done Ties that Bind which gives gives you an older sister. It's not that bad of a companion mod, not to spoil anything, but if you have FWE installed and try to do the first quest for your sister, it can be a bit difficult.
There's also TTW, which is for New Vegas. Essentially it allows you to bring in the Fallout 3 game into New Vegas, and you can travel between the two. That means you'll need a modlist for New Vegas though :)
All of those above an be found at the nexus site except for Darnified, which is on the modsreloaded site.
