Some good advice above, but here's how I'd approach things:
A pump is probably the more import shop tool, IMHO. And, good tire levers belogn on the bike as well as in the shop. Multiple sets are good. If you have more than one set, get different kinds (like one plastic, one metal).
After that:
1) I'd go first for a good set of metric allen wrenches. Not the ball-end kind either (because they can slip). Almost everything except derailleurs can be adjusted with these. BTW, sometimes, on less-expensive bikes, you may find some fractional inch needs, so a second set of allen wrenches can be a good investment
2) A good set of screwdrivers would be next - so you can adjust the limit screws on the derailleurs. You could just get a couple of small phillips and straight blade screwdrivers, but Sears regularly has Craftsman screwdriver sets on sales for next to nothing. And they are useful all over your household.
3) Moving on is a small adjustable wrench - 3" or 75mm. These can fit in a number of places, and have a number of uses - even unscrewing a valve core.
4) A non-marring hammer is really useful (not ball-peen unless you're doing metal manipulations - which isn't typically part of bike mechanics). I prefer a rubber head as my 1st choice. Ultimately, you may want multiple types sizes.
This get's me through basic adjustments.
5) Cables & housings need regular maintenance, and here we start to look at bike specific tools. A cable cutter is nice, but it should be bike specific, with the ability to also cut housing and crimp ferrules.
6) A 3" needle file and a 6" flat file, 2nd cut (that is about medium coarseness). This to clean up the ends of the housing after cutting.
7) You really need a cable puller to properly install cables. This too is a specialized tool, and can't be had cheaply at Sears or Home Despot. Nice ones have a lock position, but I have one without and have used it for years.
8) My next in line would be a chain tool. Size according to the number of cogs on the rear of your chain. Park tend to be good, and have good availability of replacement pins (which you will need eventually).
9) Depending on your pedals, a pedal wrench might be warranted. However, I've freed tight pedals with a good (Park) cone wrench and my rubber hammer. And most pedals today use an allen wrench (either exclusively, or optionally). It can be hard, but I've done fine with a regular allen wrench, not a long-armed pedal allen wrench. So look closely at your pedals and use your judgement.
12) If you have some form of caliper brakes, you'll want a wrench to center same. Could be a brake wrench (or set) which shouldn't be too dear (although mine is 40 years old, so I can't claim to have been in the market for a while). Also a suitable cone wrench may be found to fit, which also shouldn't run much.
13) Check to see if your hubs used cone and cup bearings, or some form of cartridge or sealed. If the former, bring them to a shop and select cone wrenches to fit. Remember, you'll probably want one of the same size for each end of the axle. If the bearings are cartridge or sealed, you're probably better off leaving their maintenance to a shop (although pre-tension should be adjustable with your allen wrenches).
By now, you're able to handle much of what can/needs to be maintained on your bike.
13) A shop stand is handy, but unless they're heavy and expensive, they are mostly for tuning. You can't generally bear down hard on a soft stand. If you have a workbench, a 4" or larger bench vise is nice too.
14) As you expand your tool connection the collection, consider a spline tool to remove/install the rear cassette (which is a wear item, sooner or later).
15) Then look at BB tools (and crank puller if necessary) to fit your bike.
Once you've covered this much ground, there are lots more tools, but most aren't appropriate for home shops, unless you have plenty of money, or need to do lots of more complicated work.
Hope this helps.