So, like many people of my ilk I’ve had a gmail account since way back when you needed an invitation to join. (You actually might still need an invitation to join, I have not tried to create a new account recently.)
I’ve also had my own domain for more than 10 years now. It used to host its own mail, but after much gnashing of teeth, and the ability to transfer being made nearly transparent, I converted my domain hosted email to a Google Apps account. It worked flawlessly and I have been happy for many, many years.
…and then Google decided that it wanted it’s Google Apps accounts to have more of the functionality and features normally saved for it’s own full Google accounts. A noble desire, for sure, but it came with it problems of it’s own. For you see, Google accounts and Google Apps accounts used separate authentication structures. This allowed for one to be logged into both (or more) accounts at the same time. However, upon conversion, the Google Apps accounts now use the *same* authentication cookie as the Google accounts do. You can not log into both accounts at the same time, unless you use two different browsers.
So, option 1 is: Use two separate browsers to log into both accounts. While I am perfectly capable of doing that (and I use multiple browsers as it is for work), mentally, I was not willing to make that sacrifice to my normal workflow and have Google open in two different browsers (it used to be two tabs next to each other in Chrome.) There is some good news, though.
There is an option 2: You can configure your two accounts to allow for “multiple sign-ins”. What this does is both accounts are logged in (in the same tab even), but you’re only looking at content for one account at a time. So you can be looking at your mail, and then hit the account drop down in the top left and be looking at the mail for your other account. There are some caveats, but it’s all detailed here:
Using multiple accounts simultaneously. I’m using it now and so far I have no complaints.