Can you provide a link to the store?
In any event, painting wood is easy if you just want to whack a color on it, but quite hard if you are looking for a smooth or glossy finish. In general people use spray paint rather than brushes and then you deal with overspray (too much paint), dust etc. The general advice is to spray a couple of coats, then sand with a fine sandpaper until it is smooth. In practice this can be really challenging because you tend to sand through the paint. It’s also very time consuming. So it depends on how you want it to look/feel.
EDIT: I found the foehammer site. That’s a pretty cool joystick. I assume you have the unfinished wood version of the stealth. As they recommend on the site I would sand it (I suspect it’s sanded already), prime it, paint it with a couple of coats and then the hardest part is getting the lacquer or urethane on it and smooth. The more time you spend on this the better it will be. If you just want something colored, then you can probably knock it out in a few hours. But lacquering for a nice clear coat is one of those things that takes hours and you have to keep adding coats when the other coat is dry but not set. You for a stick like this you might need a couple of cans of spray lacquer, and if you want to get it smooth like they do at the FoeHammer store you will likely need to wet sand it. If you are really interested in that look up guides to guitar painting (super similar idea) and maybe even practice on some scrap wood to see if you like what you get.
EDIT 2: If you have an auto paint shop nearby or know somebody you might ask them if they can spray it for you…