Well, I think you are not really talking about "an Internet connection", but "a high-speed Internet connection".
I thought we all knew how the Internet works by now - it's a series of tubes! :)
Actually, that's a pretty accurate description - a series of tubes, of all sizes, shapes, and types. Every possible transmission media - from copper wires to optical fibre cables, to radio and satellite transmission, to (at least experimentally) carrier pigeons.
The Internet isn't limited to any one transmission media, nor is (what you are really interested in in this case) end-user connectivity, often referred-to as "the last mile".
You certainly will be able to get dial-up service over your (not yet installed) phone line. Of course, dial-up service is slow.
Other possibilities are:
DSL, which is delivered over a copper pair - sometimes your existing phone line, sometimes an additional copper pair. (In most places, the phone company has already placed multiple copper pairs in each dwelling - they just have to be hooked-up at the central office.)
ISDN, an older technology also delivered over copper pairs, but still important in some parts of the world.
Cable - delivered over your cable TV connection.
Wireless - in some places, Internet service can be delivered to residences wirelessly. There are several variations on this, from point-to-point services where you have to install an antenna pointed at a specific tower, to city-wide Wi-Fi installations, to "borrowing" your neighbors Internet connection through their open Wi-Fi node.
Satellite - the ultimate fallback option. As you're not in the Australian outback, you're unlikely to need this.
You probably just need to ask your neighbors what is available. :)
What services are available and who you buy them from will depend a lot of local telecommunications regulations. In the U.S., for example, the telephone companies must make their copper pairs available to third-parties wanting to offer DSL service. In other places, your only choice might be a government-monopoly phone company.