[Not intended to be a complete list; trying to leave out stuff already mentioned unless I’m adding something.]
What tools you need is absolutely dependent on how you build your robot. If you’re starting with a recent KoP chassis, one of AndyMark’s toolkits is appropriate, as they include the proper Allen and hex drives; a way to cut the chassis down to your requirements is also necessary. If you decide to do a VersaFrame chassis or do VF manipulators, a hack saw and hand riveter that supports 5/32" blind rivets is essential, and a miter box strongly recommended.
In general, the more you standardize your fasteners, the fewer tools you will need. Note that many of the standard FRC parts require specific tools; standardizing on these is great for reducing your toolbox, but may not be the least expensive solution overall, as 10-32 allen cap screws are generally several times as expensive as screwdriver-driven 10-24 machine screws. On 3946, we eschew 10-24 in favor of 10-32, but otherwise favor coarse SAE threads. For certain motor mounts (e.g. NeveRest series), there is no alternative but metric threads.
You will also need some basic electrical tools - a wago tool (or small stout straight-blade screwdriver) to insert/remove wires from the PDP, several pairs of pliers (cutting, gripping, needle nose, stripping, crimping), and the allen wrench for the PDP main connector (5mm?) are essential. As with mechanical, what additional tools you will need is determined by how you wire your robot. You could get a number of crimping tools, a soldering iron, or both.
Metal/wood stock cutting: A miter box with wood-cutting back saw and hack saw that fits the miter box is the minimum; I recommend adding a couple of miter/bevel saws, one with a wood cutting blade, and another with a non-ferrous metal (aluminum) cutting blade. As an intermediate (or additional) stage, a portable band saw with a metal-capable blade is awesome. OBTW, when you make bumpers, pool noodles respond awesomely to a sharp back saw in a miter box.
Some tools which are not strictly essential but highly useful for the cost are a heat gun (shrinking insulating tubing, working polycarbonate, sealing the end of ropes), a cordless rechargeable drill/screwdriver or three, and a small Dremel or similar tool set. Try to keep the Dremel tool as the last refuge, but there are so many cases (especially at competition) where it can get in and do the job so much more efficiently than anything else that it just doesn’t make sense not to have one.
Going back to the utterly generic and not strictly necessary, I’d absolutely recommend investing in a moderate size mechanics tool kit, with combination wrenches, ratchet/socket set, allen wrenches, (all SAE + metric) and a variety of screwdrivers. There is just such a wide application of this toolset that I can’t really imagine a robotics team (or for that matter, my car) not having it.