You need to find a hospital lab in your area that has a microbiology lab or some other place where you can request some expired blood agar plates. Then you will also need some place where there is a bunsen burner and metal loops that you can use to spread the bacteria. Then you will need a warm place to incubate your plates - I suggest the top of a refrigerator (freezer on top kind).
To do the study, you should have more than just yourself as a subject and you also need controls (check with your science teacher about the rules). I think from your description, you are planning on trying one treatment, then the other.
One thing I remember from a similar mouthwash study was that after the subjects quit using the mouthwash, the number of bacteria was actually higher in their mouths than prior to starting the treatment. Thus, you would have to wait some weeks between treatments to compare.
I would suggest NOT using saliva as the bacteria in saliva are vastly different from those on your teeth and the ones in saliva do not cause the cavities. After using the toothpaste, wait a few hours or first thing in the morning before brushing, then swab the teeth! The swab is spread in a small place on the agar plate and then the metal loops are sterilized in the flame of the bunsen burner, cooled, then used to spread the bacteria over the whole plate. If you find a place to provide you with plates, they should be able to show you how to plate the swab and spread it out.
Once you have inclubated the plates, you need to roughly count the numbers and types of bacteria that you see. I will give you another hint...you need to have another control to compare your two types of toothpaste with...brushing without any toothpaste. Interestingly the bacteria that stick to your teeth will not cause cavities if you simply brush them away, although, brushing alone is not always enough either. If you go to a library, you should be able to find a good book on "diagnostic bacteriology" to help you and you can find some great web sites on teeth and a newish topic called biofilms. Check out this website about analysing biofilms. I recommend this site as the director, Dr. Costerton is one of the earliest pioneers in the subject of biofilms and he is Canadian. He used to be at the University of Calgary where I am. From Costerton's page here at science.ca you can visit his homepage. There's also a good picture of biofilm on teeth.
Mandy, from the Internet asks: What is the agar in petri dishes made of?
Anwer: Agar is a polysacharide which means it's a kind of long chained sugar. Like jelly, but it's not exactly sweet. It's used in petri dishes to grow bacteria because it resists digestion by the enzymes produced by growing bacteria. It stays firm, unlike gelatin, which turns to liquid when it is digested by enzymes. Find out more by reading about agar at Wikipedia.
What you need is a petri dish! A petri dish is a round dish filled with a nutritious agar substance that bacteria and mold love to grow on. Check this science fair website for petri dish experiment kits.
Prepare the petri dishes very carefully, according to directions. Use sterile methods the best that you can so that you don't end up growing environmental bacteria. Make 3 or 4 plates for every condition you want to test. This is good scientific practice. Keep them in the fridge wrapped in plastic wrap when you're not using them so that they stay sterile and clean. Then, you could try making a very thin layer of toothpaste on the surface of a couple of plates. Make sure you mark which plates have toothpaste on them. Take cotton swabs, and swab the inside of your cheek. Rub the cotton swab gently over the surface of the prepared petri dishes. Then, give the bacteria time to grow. Bacteria usually double in about 20 minutes at body temp, and much more slowly at room temp, so if you leave them on a table, by the next day or two you should start to see colonies growing. If your hypothesis is correct, the plate with the toothpaste should grow less bacteria. If it doesn't, then you might need to find an explanation for that! Once the plates have grown to the point you are satisfied with, keep them in the fridge until your project is done. You might want to put a bit of bleach onto each plate to kill the bacteria before you throw them away.