The connections are fine, at least in principle. I do not know the condition (or brand) of your track. Doesn't matter. It should work.
You don't need multiple transformers.
One of the most valuable tools that a model railroader (of any kind) can have - indeed, it should be required with your first traxk purchase! - is a multimeter. They can be had a Harbor Freight (etc, etc) for 7 bucks, sometimes.
This allows you to, among other things, check:
1 - voltage, AC and DC; you need to know this, sometimes.
2 - continuity - that is, do we actually have a circuit? (as on a piece of apparently dead track...)
3 - can be handy around the house, in general
4 - just read the instructions
