First of all, Congratulations, you've come to the right place. This is a community where you can find everything related to competitive tf2 if you ask the right questions.
There are basically two formats of tf2 that are considered to be competitive: 6v6 and Highlander (or 9v9). They can both be played in the regular maps that you're used to play in pubs, but they differ in the classes restrictions. 6v6 is played with 2 scouts, 2 soldiers, 1 demo and 1 medic per team, and 9v9 is played with one of each class per team. What you've been playing in lobbies were highlanders, and they're considered to be less fast paced and more disorganized (in low levels of competition) than 6v6, whereas 6v6 is very skill demanding and a single death of someone in the team might be very important for the outcome of a round.
For starters you should keep playing lobbies, that's what I used to do when I found out about comp tf2; eventually you'll find out that you're not going anywhere playing them and then is when you should move on to newbiemixes, tf2mix, tf2.pug.nahl and tf2.pug.na. You can find all the info about those in this site somewhere. Also, if you see the right sidebar of the site there's a tab titled Active Servers, you will find there some very cool servers where you can play with other people from this community and improve your skill by playing in DM and MGE servers.
Wish you luck, this is a good read btw:
DrakeMegrimhttp://teamfortress.tv/articles/view/52-introduction-to-competitive-tf2-part-1
Edit: You can also watch streams, there's usually people willing to answer your questions