Best DIY all-rounder cutting tool for straight edges? Multi tool?
The problem with 'All-rounder' tools is that they are rarely actually very good at anything. Sure they'll let you cut a wavy line in tile, but how often will you be doing this?

My advice would be to buy good-quality tools as, and when, you need them. An SDS drill, battery drill-driver, and decent hand-saw will get you a long way. If you'll be cutting sheet material, then a circular saw would be useful. Later on, you might want to get a mitre saw, jigsaw and a router.

One point on buying power tools: With the exception of the drill/driver - don't get cordless, unless you have a very good reason; When the batteries die, they are expensive to replace, if you can even get replacements, although this is less of a problem nowadays as Li-Ion batteries typically last much longer than NiMH / NiCad. 
As noted in the comments, they are also often less powerful than their corded cousins, unless you pay a lot of money.

For your requirements: 


Wood - hand saw
Plasterboard - Stanley Knife for straight cuts, padsaw for curves
Chasing wall - SDS drill with chasing chisel bit