The most important thing you should buy is a good test kit. The one we recommend is the Taylor K2006 or 2006C which you can get through the test kit link in my signature below. (You won't be able to buy it locally.) Whether you can use a floater with trichlor tabs (and some of the other products) depends on your testing results once you get the good test kit. The test kit is a little pricier than some you can buy but is definitely worth buying. It will make your pool maintenance much easier. (And, with a new baby, easy is good, right?)

--You don't need clarifier.
--The Arm & Hammer is just expensive baking soda with clarifier added. You are better off to just buy regular baking soda. Again, you don't need clarifier.
--The algaecide is ok but you actually don't need algaecide if you just maintain proper chemistry. I never use algaecide in my pool.
--You will need a brush. That one is fine.
--Whether or not you can use the shock depends on your water testing results. Once you get the good kit and provide results, we can help you determine whether or not you can use it.
--Same for the chlorinating tablets. Depend on test results.

Get a kit and then post some good numbers and someone here can help you go from there. Also, when you post your numbers, tell us what kind of size of filter you have and how big of a pump you have.

As far as searching, until your registration is completed, you'll have to log out first to see the rest of the forum. Also, instead of using the forum software's search feature, it is better to use the one that is linked in either mine of PoolDoc's signatures. It will give you a lot better results.

Welcome to the Pool Forum!