Effects and Amp for Sax
Adding effects through amplification is tough in a small space like a bed room because the sound loops back through the mic and you soon get distorted noise and feedback. As noted by others the small space is great for practicing to create the vibrato and other effects available through playing style and choices in mouthpieces, and reeds.
As you add FX and amplification you need to consider three pieces. 1) The mic. 2) the effects box, 3) the amp. If you go this route you have to have a way to carry all this stuff and find power when you want to use it. Much of the reliable effects equipment is designed for rack mounting in road boxes and is not small.
1) Condenser Mics provide better coverage of all the nuances and overtones which you can create on the sax but they require phantom power and careful location. The power is provided by the first "box" the mic is plugged into. If your effects box does not have phantom power you need a small mixer or phantom supply. The physics of mics vary, but most others which do not require power are dynamic microphones. The condenser mics are a more costly than dynamic. The condenser mics can be more sensitive to location and more susceptible to feedback. A reasonably priced (under $100) option for a full spectrum dynamic instrument mic is Shure SM57. Good clip-on bell mics are available usually for significantly more money. Most are condensers and come with there own dedicated power supplies.
2) The effects depends on the Mic input and need for phantom power. I have not found a small unit with phantom power. I can recommend Alesis effects processors and for small size the Alesis NanoVerb should work fine. This is primarily a reverb unit so if you are looking for more distorive effects you will have to do more research. I have one freind who plays through a guitar foot peddle which as reverb and other distortions as well.
3) For the amp you have to consider if you will ever play outside the bedroom and how you may connect to a sound board for performance. I use a Roland Cube 30 which is very small and quite powerful. This is a full spectrum piano amp so it does not limit outputs like guitar or base amps. It accepts multiple inputs including balanced Mic and 1/4 inch. On-stage you can set it up as a personal monitor and the outputs can be used to feed a larger sound system. I strongly recommend against using guitar or base amps for instrumentals.
Conclusion:
All the electronics can be fun and costly but in the end they CAN NOT make you a better player. Amplification only makes things louder; the bad with the good, and effects are really distortion of the true sound of any instrument.
For the whole fun effects sound I use the Roland Cube 30 and Yamaha Wind Midi (over 400 sounds to play with). For Alto flute which does not project well, I use the SM57 and the Cube 30. I found there to be too many boxes and power cords to add phantom power and effects. (Multiple inputs on the cube let me set-up the Wind Midi and the instrument mic at the same time). To hear effects added by house mixing I use a dedicated monitor circuit and trust the sound tech to set effects to match/blend the band and vocals.
Play-On