If you use a playlist (even on shuffle), you can press play on your computer and the music will stream to the speakers. This method is cheap and easy to setup. If you want to pick and choose songs you need a remote to control the computer (like iPhone controlling iTunes on computer) or a dedicated media player. It sounds like you are looking at either the remote or dedicated player option, with each having their pros and cons.
The classic iPods (pre iPod touch) are ancient nowadays (but work well for music although not all had Bluetooth) and the iPod touch is always using old hardware (the recently updated model in March comes with a iPhone 7 processor which will be 3 years old with the new iPhones in September). So if you were to go the iPod route then keep in mind the device is underpowered out of the box (and check for Bluetooth). For music it will be fine but it doesn't run apps like an iPhone.
For pure music playback it shouldn’t matter which device is used, as both can either use the built-in music player or get a few music apps. This is why I mentioned using a iPad for the home, as you get a device that can play music and use a pencil for drawing and a larger screen for reading or watching. If you go the phone route there are many phones with decent cameras that you can also use as a point and shoot camera on trips. The iPhone SE is only a touch slower than the newest iPod touch and it has a better camera. The recent and cheaper Pixel 3a has a very impressive camera that compares well against the newer iPhones.
You really should start with a budget and choose what your device should do. If it is just music then anything will work. The only limitation is music library size. Android phones can (sometimes) use a microSD card for extra storage (the recent models can support up to a terabyte card) and a computer can always add a hard drive. If you want a dedicated device, then see if your music library will fit.
For purchases, eBay is good as they give a bit of consumer protection. Keep in mind that if the device is old it might not have the best battery and might need to always be plugged in. Battery replacement requires a steady hand and is a delicate operation at the best of times so factor in the cost of a replacement service if the device is old and you want to use it away from the charger.
Hope this helps.