I've found that there are two big dampers on reading comic books cost and finding complete stories.
When I started reading DC comics I went to a used bookstore and looked for consecutive issues of individual titles. I wound up with a 5 issues of Teen Titans from 1995 and the first 3 issues of Young Justice for around $10. That said I was able to read large plot arcs without getting too lost.
I often still use this tactic to find new (always used) comics for my collection. I look for titles from a specific year and fill in runs of an individual title. Sometime special editions are a great place to go to. For 99 cents I was able to get a single part issue of the Atom which was pretty complete and easy to follow (And like 4 times as big as a normal comic book).
Also graphic novels and/or volumes containing multiple issues of a story arch can be a great and budget friendly option as well.
You mentioned liking 3 of the biggies in the DC universe so while you are familiar with the big names, some of the most fun characters are a bit lesser known. You might want to look around and find out what other titles might interest you. A good way to find that out is watching cartoons and seeing which "guest" characters make an appearance and finding out via the internet what titles and issues that particular character makes an appearance in.
I'm not much of a fan of the new 52 as so many of the characters I really like got dumped or had their origin story butchered by the reboot, but you might like it and have an easy time acquiring consecutive issues with it though.
Also, wiki pages tend to have good records of issues pertaining to various character arcs, so if you want to find out the origins of a particular character (as it may not be shown in issue #1...probably won't be shown in issue #1) the source materials are usually linked to that bio.
The best advise I can give you is find the titles you truly find interesting and have fun.