Personally, I'd suggest selling everything you own, throwing a dart at a map, going there, and proceeding to see where adventure takes you. When you run out of money, find something productive to do and start earning some more money.
$20k won't get you very far in financial terms, but it can buy a hell of a lot of life experience from the position you're in. The cost of long term travel will rise rapidly as your life continues; right now that $20k will probably buy you a year, but in another decade, a year of travel will be about as feasible as a trip to the moon.
You could actually do this for less than $40k if you had a co-founder. This guy just paid retail price for a developer and designer but if you either knew how to do it or got a co-founder you could do the same thing for less than $5k probably.
"... and wants to grow rather than blow the amount."
A startup is a great way to blow the whole amount, unless you are pretty sure that there's an actual market for what your startup will produce, that you can produce it, and that you can win over anyone else who's also trying to produce that. Many, many, many startups do not fit those criteria.
If I was in your shoes I'd probably buy myself 1 nice thing (a small vacation or a new piece of furniture/electronics or something) and I'd split the rest between investing and having an emergency fund for when life throws you a curve ball (home/car repairs, losing job, illness, bail money, etc)
How does one do this? Does it involve a broker? Is their cut irrespective of market performance? Do the anecdotes and unspoken secrets about "mom-and-pop investors" not apply?
$20k won't get you very far in financial terms, but it can buy a hell of a lot of life experience from the position you're in. The cost of long term travel will rise rapidly as your life continues; right now that $20k will probably buy you a year, but in another decade, a year of travel will be about as feasible as a trip to the moon.