Can you buy a new phone and put the old sim card in it, even if it is not the same brand of phone?

Yes, as long as the new phone will support sim cards.

Now, if you have switched the phone service company (e.g. - from tmobile to at&t), old sim card won't work.

You can put any sim crd in any phone as long as its open to all net works xx twinkel xx :D