Tools completely depend on the project, and you can do a surprising amount with only hand tools (example: houses built much before the 20th century weren't built using power tools).
For general demolition, I find a rip hammer (straight-claw claw hammer) and an 18-24" crow bar to be useful tools, along with a good set of work gloves and a reciprocating saw.
For construction, you need good layout tools (tape measure, square, level, pencil, plus maybe a chalk line and straight edge), plus the tools to cut your materials to size and the tools to drive the fasteners. That could be as simple as a hammer and a hand saw. Sawhorses are also useful things to have, but you can often build or improvise them. And if your layout isn't perfect, a deadblow hammer is useful to make things fit, as is a set of chisels and a block plane.
For general painting, you'll want a putty knife, a 2" sash brush, a medium (9") paint roller with cover for the surface to be painted, and a roller tray.
The power tools that get the most use for construction are the cordless drill, cordless impact driver, miter saw, and circular saw. Some will argue for the table saw, as well, A pneumatic nail gun and air compressor are significant time savers but perhaps not worth the cost if you have more time than budget.