You can actually get a high quality stereo integrated amp for $200 or less, however very few have a sub preamp output - most/many will have stereo preamp outputs though and that can be used to feed a sub that has stereo line level inputs. You will have to shop around for deals though. If you can find a good used stereo receiver or even a 5.1 receiver though, that would work as well and likely cost even less. The integrated I use in my bedroom was around $700 new and it goes for around $200-300 used.
For the large speaker, you theoretically use it as a sub - open it up, disconnect the other drivers and use it just like that without building anything (may want to do some tweaks to the enclosure though, if needed). This will need a dedicated amp though.