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The question has such a wide scope, the answer could require books to be written. Robotics is a huge multidisciplinary topic involving lots of types of electronics, mechanics, software and can involve just about any other kind of technology as well.
First you will need to decide on the scope and objectives - things like:
Next you need to decide how much you will design versus how much you will buy
Next learn whatever you need to design the system and the parts you want to design
Then figure out what tools and facilities you need to build your robot and get them together.
You will need a bunch of small hand tools, a voltmeter and a computer to do the programming.
Don't get too hung up on all the preliminary stuff - get to this point quickly and start building, because you will learn as you go and realize you wasted a lot of time on uninformed preliminary decisions.
A good way to get started is to buy a kit that you can afford but that stretches your technical skills to a new level, but not so hard that you will get discouraged and give up.
When you get interested in some aspect of the robot, you can focus on custom designing / replacing that aspect.
What are the most useful things when building a robot?
Resources. No, that's not sarcasm! When my school (at which I taught Design & Technology) entered the F.I.R.S.T competition they sent us a large plastic box containing a lot of parts, manufactured components, many of which I'd never have thought of some of which I didn't even know about. For example, an automobile electric seat adjustment mechanism, the Fisher Price drive unit that they use for various childrens electric vehicles and many more similar pieces. I'm sure these goody box contents must be on record somewhere but they were enough for us to build our robot without additional components being needed (which was good as we were flat broke!) a factor common to many an inventor/prototyper, I'm sure.
I've always relied upon junkshops and fleamarkets and discarded consumer products left out for trash to source most of the stuff I need and very often when considering a robotics project it might end up being shaped as much by what was available as whatever ideas I had in the first place. In fact my best projects were all shaped that way, I'd take some toy apart and what it revealed led me to redesign accordingly and usually resulted in something better than the original idea.
Toys like Meccano (Erector) and LEGO are worth their weight in gold when prototyping and they're expensive to buy new but fleamarkets sometimes have job lots of broken up sets for sale dirt cheap which is a good way to acquire the gears, shafts and so on that would otherwise be expensive to purchase as needed. Many a wire or radio controlled model vehicle can supply a wealth of useful ready made sub assemblies and once in a while a scrap yard can supply real treasure like the time I found some old mainframes which supplied me with dozens of high quality relays not to mention the wiring, motors and so forth.
Yes, resources are the most essential thing to source when a robotics project is on the drawing board.