Are there any quick way to tell if a filter is high-pass, low-pass, or band-pass, just by looking at the transfer function in the s domain?
Yes. Evaluate the function as s approaches zero and as s approaches infinity. That will give you a very quick look at low and high pass filters. Band pass can be a little trickier, and may require some factoring first to get it to a form that makes sense to apply the aforementioned process.