Ballet classes are fine alongside your ballroom. There are certain concepts that all disciplined forms of dance share in common. However, you may find that as time goes on and you start spending more and more time working on your ballroom, you will have less and less time to take ballet classes. In the meantime, if you have extra time, it can't hurt to study other forms of dancing.
The typical way to go about studying ballroom is to find a partner with similar goals, and take lessons and practice together. A typical practice schedule for a competitive pro or amateur couple is anywhere between 6 hours per week (eg 2 hours, 3 days per week) and 18 hours per week (eg 3 hours, 6 days per week). Start with fewer hours and work your way up as time goes on... in the beginning, you won't even know what to practice after the first few hours. Later on, no matter how many hours you have, they'll never seem like enough.
As for lessons, find at least one reputable teacher you trust, and take 4-6 hours per month of lessons. Again, less in the beginning is fine, especially if you are working on a limited budget. But you'll want to work up to that number ultimately. Private lessons are expensive, but extremely effective if you have a good teacher. And you you're only really paying half if you have a partner to split the cost with.
You can take with more than one teacher, although you should try not to spread yourself too thin. Most people find that one or two core teachers are ideal, with occasional lessons with visiting coaches for a fresh perspective. As they say, too many cooks can spoil the broth, so limit your primary teachers to one or two (or perhaps as many as three, if you don't find yourself being pulled into too many different directions). In the very beginning, try a few different teachers out, so that you give yourself a sense of what type of teacher is most effective for you. Then choose the one or two you like the best.
If you can't find a partner right away, don't let that stop you from taking lessons. You can begin your search for teachers anytime, with or without a partner. If you're on a budget, you'll have to take fewer lessons per month until you do find a partner, but you should do it anyway. A partner probably won't fall into your lap if you just sit around waiting for one. More typically, you'll find that it happens when you least expect it, while in the middle of focusing on your own dancing. Also, the better you get, the more you increase your odds of impressing the prospects. So take as many lessons as you can afford in the meantime.
If you do have a reasonably large budget, you might also consider doing pro-am competitions while you're in the process of finding a partner. This will also help you improve your craft in the meantime. It also helps increase your exposure to prospective partners. But it is expensive, so don't do it unless you can comfortably afford to.
As for specific teachers and studios, I don't do recommendations. But let us know what city you live in, and perhaps somebody here will have an idea. You can also check out our directory at
https://www.ballroomdancers.com/Directory/ for a list of studios and teachers in your area.
To get started with your search for a partner, take a look at the listings in our classifieds section. You might even want to post your own ad.
https://www.ballroomdancers.com/Classifieds/.
Good luck with your career!
Regards,
Jonathan Atkinson