As a general resource, Park Tool has a great collection of service and repair guides. They list the tools you need for each type of repair.

Here’s the basic maintenance checklist for an old neglected bike:

• Inspect anything rubber, and replace as needed. Tires and brake pads harden over time.
• Drip some thin lube (Tri-Flow or light oil) into the cable housings. Run the cables through the housings to spread the lube throughout.
• Lubricate all the pivot points on the derailleur(s), as well as the axles of the rear derailleur pulleys.
• Lubricate the pivot points on the brakes. Be careful not to get lube on the brake pads.
• Lube the chain and check chain “stretch” using a chain gauge. Or just replace it and start fresh.

And if you really want to get into it, here are some more advanced maintenance items:

• Repack the wheel hubs with fresh grease.
• True the wheels.

To build a basic toolkit, start with this:

• Tire levers
• Phillips screwdriver, size 0 or 1.
• A small flat-head screwdriver.
• Metric hex/allen wrenches. 4mm and 5mm sizes are the most common. Get a set that goes from 1.5mm to 10mm. Having ball ends on the 4mm and 5mm wrenches can be a lifesaver.
• Metric box-end wrenches. 8mm, 10mm, 13mm, 15mm, 17mm are all common sizes.

More advanced maintenance will require special tools specific to the parts installed on your bike. The designs of freehubs/cassettes/cranks/bottom brackets have all changed a lot over the last few decades.