Some have suggested I simply buy a tape-to-CD dubbing deck, but they are cumbersome and the last thing I want is a shelf of separate components. Right now, I don't have any audio components, not even a portable. Cost is also a factor (keeping it low as possible).
It sounds like you want an all-in-one audio system with cassette-to-CD dubbing capabilities, but I know of no such product. A dubbing cassette player/CD recorder would be the simplest solution, but as you point out, you'd also need an audio system to hear what you're doing.
Do you have a computer? If so, I recommend getting a good cassette deck, USB audio interface, and audio software, which will let you convert your cassettes to digital files in the computer. This is the most flexible solution, because you can digitize the cassettes at 44.1kHz/16 bits uncompressed, which are the specs used by CDs, and most audio programs include various editing capabilities to reduce noise and otherwise enhance the sound quality. Then, you can burn CDs or convert the files to MP3s as needed. But be sure to keep an uncompressed copy of the files for future use.
Examples of the software you need include Acoustica Spin It Again, PolderbitS Sound Recorder & Editor (seen here), and Digitope Cassette to CD and MP3. These programs and others like them can work with the analog audio input on the computer, but you'll get better sound quality if you use an outboard USB audio interface. Examples include the Numark Stereo iO, Behringer UCA202, and Ion Audio Record 2 PC.
Neither the software nor the audio interface is expensiveeach of these examples is less than $50. A good cassette deck might cost a bit more, but aren't your precious tapes worth a little investment?
If you have an A/V question, please send it to firstname.lastname@example.org.