These days, tape pieces are relatively low-tech. Just about everywhere you go there is at least some sort of PA system with a CD player. At the very least there will be a stereo system. I've chosen to go the route where I bring everything along with me but the amp and speakers, though many others simply bring a CD that they can provide the sound guy. The reason for my preference is simply that I want to be able to control when the tape starts, to control the balance between the bass clarinet and the tape (I use a microphone -- more on that later) and the volume of my monitor.
A bare minimum: CD or iPod and cables to plug it in. The cable you'll need is one with an 1/8" plug at one end (for the ipod) and a stereo RCA (or better, 1/4" plug) on the other. Most venues don't have the adapters. This cable is less than $10 at Radio Shack. Total cost: $10 or less.
From there, you might want to have a decent microphone. Why? In order to mix your sound with the CD/iPod, you need to amplify yourself. That way, you can boost the tape part without drowning out the live part. It also blends your acoustic sound with the tape before it goes into the hall. It should be one of your first purchases. My personal favorite is the bass clarinet model form AMT (www.appliedmic.com). Those run about $800, but there are others, including stationary mics (i.e. not clip-ons) which are less expensive. Fortunately great microphones -- especially ones built in China like SE Electronics -- are amazingly cheap: less than $500. Many are even cheaper. But you'll want to get one that has a "cardioid" or "hypercardioid" pattern, since those are more appropriate for live contexts; omni pattern microphones will feed back much more easily. You might also need a cable and a mic stand, though those are ubiquitous in most venues.
Microphones that attach to the bass clarinet are, for me, more appropriate because I tend to be more, uh, "active" onstage. Since I stand when I play solo, I don't want to have to be tethered to one location because of the position of a stationary microphone. Just something to think about for yourself: do you want to always stand in the same place, or do you want the freedom to move about - say if you're playing a piece which requires multiple stands and you need to progress from left to right.
A small mixer is becoming less important, especially if you are planning to purchase a computer and audio interface, since much of the mixing can happen inside the computer nowadays. If you choose to get a small mixer, a Mackie 12-channel ($369) is the one of the more popular. They are inexpensive, sound good and are built like a tank. If you choose to go smaller and a bit lower quality, you can get one from Behringer or Alesis for as low as $100. You would plug the mic and CD player into the mixer, which would plug into the PA system. You could add a small monitor so that you can hear the tape part better. A good monitor should be small, light and self-powered. The Yamaha MSP5 is great, but heavy. ($250). Another choice is the Yamaha MS101 ($130) which is much lighter but is not as powerful. Often a venue will have a dedicated monitor for you to use, but just as often, they don't (or it sounds dreadful). For me, it's better to be on the safe side.
Summary (these are minimums):
CD player or iPod: $100-200 (but you probably already have one, right?)
So, these are the basics.
If you want to go all-out, check out the main site to see what my current rig looks like.