Well, if you are interested in a beardie you will need at least a 50 gallon long tank. Allot of us have 4' long x 2' wide x 2' tall enclosures for them. You can purchase a very nice beardie and a tank, lights, and decorations for what you are looking to spend. At 13, I would recommend a beardie that is at least 4 months old. Baby beardie's are very delicate and can get sick and die very easily. If you get one that has a little size to it or even an adult it will make taking care of it much easier. Since you have plenty of time to prepare you may want to consider starting a bug colony ahead of time for feeding. If your parents are cool with you having bugs, you can easily breed super worms and dubia roaches. They make great beardie feeders and if you get a colony up and running before you get your beardie you will cut down on your cost of taking care of it ALLOT.
I just bought a tegu for my 16 year old son. It is a much larger lizard that requires ALLOT of space, so you would probably want to steer clear of any monitor type lizards considering your space constraints, with the exception of maybe an ackie monitor that gets about the same size as a beardie.
If you are interested in a snake I would recommend a corn snake or a ball python for a beginner. Corns and Balls come in allot of different morphs (colors) They are calmer snakes than say a milk or a king snake that are more likely to bite. Plus neither get really big, so they could be happy in a 30-40 gallon tank their whole life. Also if you are interested in a snake there are certain kinds that can be kept together. I once saw a woman bring into a petstore what looked like a ball of corn snakes. It was really cool because they were all different colors and they were just clumped together happy as clams.
Allot of people on here have geckos, they can be held daily but are much shyer than a beardie. They also don't need allot of space, but do require allot of moisture.
Turtles can also make great pets, the terrestrial kind aren't hard to take care of, but make sure you know what you are getting. Some of the tiny little guys you can buy can grow as big as a truck tire! Of course this isn't going to happen overnight but it is a consideration, some turtles can also live for a VERY long time making them a lifelong commitment.