I am going to nth the recommendation to purchase a used HF transceiver. When I got my license I took the approach you did.
However, the challenge is the easy/inexpensive radios to build are mostly CW (morse code) only. Learning CW is fun, but it's one more barrier, and will keep you away from other interesting ham radio activities (like digital modes). There are solid SSB/AM/etc transceiver kits (like the Elecraft ones someone else mentioned) but they are much more expensive.
The homebrew ones also tend to be lower power, which is fun, but will make it much more frustrating to try to make your first contacts.
A straightforward project I recently completed was to build a TNC Pi (http://tnc-x.com/TNCPi.htm). That will only require a VHF radio, and you'll get to put the kit together and (likely) make your own connector cable.
