"But I get confused about how do people get into a competition - must it be through a studio?"

It depends on if you're dancing Pro/Am or not. If you're dancing in Pro/Am events, then sometimes the studio handles the entries, sometimes the teacher, and sometimes the student. It really depends on each particular case. My teacher was an "independent" -- not affiliated with any chain or studio -- and we chose what competitions to enter ourselves and I'd handle the entry forms etc.

You can find out about when and where a lot of competitions are by checking out usabda.org and ndca.org

"what's with this bronze/silver/gold stuff?"

These correspond to syllabus levels. Most dance schools, and many teaching organizations, divide the foundation steps for each dance into different levels. Students often start by learning the Bronze steps, then the Silver ones, and then Gold. Many competitions have syllabus-restricted events where students can enter at whatever level they've been working on thus far.

"If I had a partner (and I have been looking) do we have to go to the same instructor - or can we go to private lessons alone and then work together with sometimes a coach or lesson together?"

You two can do whatever works best for you. Some couples see a coach together. Some couples keep their own private teacher and also see a coach together. You will need to see a coach together, for sure. What most people do is see a coach together once a week, and practice as much as they can together in-between, and then they'll see special coaches together from time to time for extra help.