Dial-up is definitely much slower than DSL. From what you've described, I'm not sure which you have but it sounds like DSL. Usually, a DSL connection would mean that you had a telephone line from the wall to a router or an external modem with a USB cable from the modem to the PC. However, you can get internal DSL modem cards that go inside the PC so it might be that you have one of those. You might even have an internal ISDN modem - ISDN is a 64kbps or 128kbps digital connection that's not as good as DSL but is still notably better than dial-up.
The simplest way to check what your connection speed is like is to go to
www.speedtest.net and run the download/upload test there. It's brilliant. If you get a result of over 130kbps download speed, then you have DSL.
When you are looking for a new ISP, you will probably find that the business services are twice as expensive (or more) than the residential services. In some cases, this is just because the business services have better technical support. Quite often, however, there will be additional features such as a fixed IP address as part of the package, as well as better performance and guaranteed service levels. Residential customers basically just get whatever they are given, which might be reliable or it might not - if it goes slow for a long time, you can phone the support number but they might not be very interested in making it better as it is only business customers that usually get a guarantee about how good the service will be.
These are my opinions based on my experience in the UK - other countries might be different.
