I want to create an open source custom home page and I need to use logos (such as Google's) to link to their sites. Can I use them freely, or should I check individually, and if it is the latter, how do I check, and does licensing an open source project differently affect this?
Often, these things are usually trademarked, if not copyrighted as well.
However, a lot of the larger, more major companies provide to people various guidelines that allow people to use their trademarked, but they are often covered by various restrictions. Google, as an example, has a comprehensive help center for trademark guidelines. Although these may be points, they are not definite and final. It is your responsibility to make sure that you respect Google's trademarks, and that you do not infringe on any of their rights.
Provided that you have a license or permission to use a logo/trademark within an open source project, then the logo should be fine. Open source licences often only affect source code and compiled object forms of the source code, so I don't see many clashes here.
Aside, as an alternative, you may have luck finding a few things in the Google Material Design Icons.