Do you have the instruction manual for your TV? It sounds like it is one of their rear projection CRT televisions. It also is likely that it only accepts 1080i, 480i, and (maybe) 480p. HDTV comes as 720p or 1080i, so you want to make sure that you ONLY feed the television 1080i.
Which may mean nothing to you...
Here's what you want to do: Look at the back of the TV and see if there are 3 connectors side by side that are Red, Green, and Blue. It (hopefully) will say "COMPONENT" or "HDTV" next to that input. Let us know.
If you have those inputs, then going with HD cable is a great way to get started. HD cable should have your local channels as well as a couple of premium and all HD channels available to you.
Over The Air (OTA) HD is acheived with any OTA set top box. They are sold by many manufacturers, but the real kicker is that you need an antenna, and usually that antenna should be mounted on top of your roof.
Yes, call the cable company and check to see if they offer HD cable first.
Just to make things really interesting - it is possible your TV has an integrated HD receiver, but you would have to read the owners manual to find this out. It still means you need an antenna for OTA high def, but that is all you would need. I would still call the cable company.