The cassette player you use makes a huge difference to the sound you get, yes. How much you spend on one is rather dictated by how many cassettes you want to digitise. If it's only one, then borrow one from somebody - but ideally as a minimum make sure it's a desktop model rather than a portable. If it's a lot of cassettes, then the 'best in class' machines - by a very long way - were made by Nakamichi. But these are often fearfully expensive, even this long since the format effectively expired.
It would be easier to answer this if we knew a bit more about what you are trying to do, in terms of what's on the cassettes; music is a lot harder to deal with than speech, for instance, and there are other issues too, like whether noise reduction was used, etc. And as I mentioned, if it's only one or a few, it will be a lot more cost-effective to borrow one than purchase something you're hardly ever going to use.
The other issue is the cassettes themselves. If they're old, then you have to take a few sensible precautions before playing them. In many ways this is simple; bang them down flat on a hard surface first, and then spool them from one end to the other and back again. This reduces the chances of both sticking and print-through, and is always worth doing. Also check that the little felt pad is still present behind where the play head goes; if this is missing, then you'll need a dual-capstan machine to play the tape at all, or you'll have to stick another pad in there. The problem with the pads is that the glue on them dries out, and they fall off and get lost.
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