It depends A LOT on the particular bike(s) you're working on.

General tools:

• Metric hex (Allen) wrench set.
• Metric wrench set, especially if you're working on bikes from the '70s or older.
• Screwdrivers, mostly for older (or lower-end) bikes.

Bike-specific tools:

• Cable & housing cutter.
• Chain breaker, for shortening new chains and/or rejoining old non-peened ones.
• Chain whip for removing cassettes.
• Freewheel/cassette lockring tools for the particular bike(s) you'll be working on.
• Bottom bracket tool(s) for the particular bike(s) you'll be working on.
• Crank puller if you'll be working on square taper or splined cranks.
• Pin spanner and bottom bracket lockring tool if you'll be working on cup-and-cone bottom brackets.
• Big-@$$ wrench(es) for bottom bracket fixed cups and/or lockring tools. (I find a 15" crescent to be handy.)
• Cone wrenches if you'll be servicing cup-and-cone wheel hubs.
• Headset wrench(es) if you'll be servicing threaded headsets.

...and I'm sure I'm forgetting a thing or two.

Like Retro Grouch said above, start a metric hex (Allen) wrench set is an important start. Past that, it depends on the specific bike(s) and the particular work you need to do, so add tools as needed.