By all means, use the publicly available code within the limits its license permits. Always strive to give credit back (I oftentimes add notes to where I took config bits even in my private my-eyes-only repos to have some breadcrumbs).
Remember that licensing and copyrights are kind of separate things. People own copyright to their work (unless they explicitly give it up), and licenses are the terms on which you can use their copyrighted work.
Know the basics of the OSS licenses and know which ones you can copy things from verbatim (e.g. don’t touch AGPL code unless you also use AGPL). Generally, I just keep the original license and add a note to my license file saying that e.g. this code is licensed under Apache 2.0, but some parts are MIT.
It gets somewhat murkier when you use someone's code and base yours on that. IANAL, and that's very much the legal territory. If at all possible, just reuse the original copyright and license and then derive your work (given the license allows that).
Being on the receiving side of this a few times (people using my code verbatim in their projects I stumbled upon) it leaves a bit of a sour taste in the mouth when you see your copyright header replaced with someone else's completely. Don’t do that. All the three times it happened to me, the other party was quick to remedy the situation, though (2 added the original copyright note back, 1 removed all my code). So just don’t do that. Make a habit to read that dumb tall copyright notice at the top of the file every time and you’ll quickly learn what to expect.