What should I look out for before I upgrade, and should I consider not upgrading to the newest version of macOS so I can keep certain applications running the same?
Use the following reasoning to answer the above questions. At the simplest level, this is what you need to take care of:
Prepare a list of all the 3rd party apps that you use. Make sure that it includes, at the least, the apps that you are highly dependant on for your day to day personal/professional usage. You'll need to make sure if they are supported and run fine on macOS Mojave.
Mac OS X Lion and macOS Mojave are 7 years apart. With the pace of technological innovation, quite a few apps have broken/become unsupported in the newer releases of macOS.
Prepare a list of all the 1st party (Apple) apps as well. Especially list down the apps that you are highly dependant on for personal/professional usage. Apple too has deprecated/removed some of its apps. You need to make sure that upgrading to macOS Mojave doesn't leave you hanging without some apps that you are dependant on.
Apple announced a while ago that macOS will drop support for 32-bit only apps. While this was expected to happen with macOS Mojave, 32-bit apps still run, albeit with an error message that the app may stop working in the future version of macOS. (Such apps still work with the current and most likely the last public release of macOS Mojave, i.e. 10.14.6. This will change with macOS Catalina, the upgrade expected within a couple of months).
Check if any of the apps that you use (3rd party) that are still 32-bit only.
Check the release notes for or get in touch with the developers of the 3rd party apps you are concerned about to check their compatibility status with macOS Mojave. This will better prepare you with your upgrade plan.
Would keeping a separate partition of macOS Lion be an alternative?
Yes. However, it would require some work with creating partitions and installing Mac OS X Lion and macOS Mojave side-by-side. This could be some work for a casual user.
I'd recommend you to get an external USB hard drive and install macOS Mojave on it. You can boot off of the external drive (restart MacBook and hold Option key to select boot drive), while leaving your current installation of Mac OS X Lion intact. Work with this setup for a few days until you are convinced to fully commit to macOS Mojave.
Do note that when running off an external hard drive, macOS Mojave may feel slow merely due to the fact that the OS is running off of an interface with slower speed. Don't let that make you assume that macOS Mojave is slow.
This is the best and the easiest strategy you can use to test things out and transition seamlessly.