Be a bit wary here. Some instructors are going to be all about getting you into the show ring whether you yourself have any interest in it or not. They tend to draw a group of students with the same goal as them and people who aren't competitive-minded can feel left out or 'less than'.
Riding horses is NOT about competing. That's something laid on top of riding, that some people enjoy.
What happens at shows? You get all dressed up in special clothes, clean and clip your horse and polish his feet and fancy up his mane and tail etc., and bring your A game to the contest. Which typically consists of riding around an arena either with your fellow competitors or if doing some kind of course or pattern, one at a time, for a few minutes that feels like eons, and then you either get a prize or you don't. It is very expensive, even compared to owning a horse, which is saying something.
For some folks, it is the best thing in life. For others not so much. You don't need to decide yet.
While I agree with a lot of this, I'd say it depends on the level of the show. Yes, some are very snobby and expensive. My daughter is going to start doing "real" shows this summer. But to be honest, we've talked about it and she's not sure she really wants to focus on that. However, she's already done a few "fun" shows and we are going to start off the season with an "in-house show". Those are a blast! She did a fun show last summer at the end of a week-long riding camp where she got to try barrel racing for the first time! Now, mind you, a lot of the kids weren't going faster than a trot so when she launched into a canter and spun around those barrels, you could literally hear the other parents gasp, LOL. She had a slight advantage in that she's been riding for years while some of those kids were new to riding. So she took first place, but the previous year, had gotten third place (same camp, same show) and was very happy with that. She's also done a Western pleasure "fun show" where it was more of an exhibition - no prizes were given, well, except for the pie-eating contest prize which sits on her dresser :)
Bottom line is that for a lot of people, shows are a way to set concrete goals for yourself. They can be a lot of fun, social interaction and just a great way to spend a day around horses, but they can also be uptight and competitive.
I agree with Ava, however, that if your instructor is pushing to show already, that may be cause for concern IF that's not what you want. My daughter's second coach (we went through 5 in 4 years) was very much about showing. When I told her my daughter wasn't ready to show yet (she was 7!), she lost interest in her and focused on the others to her detriment. So we moved on. With the coach we have now, we had to ASK if my daughter was ready to show this summer. There was never any pressure either way.
Anyway, that's my 2 cents. Let us know how things go for you!