Your machine can run any OS from Snow Leopard (OS 10.6.8) to the latest version of High Sierra, OS 10.13.6. However, it might be wise to upgrade to an "in between" Mac OS. Maybe start with OS 10.10.5, Yosemite. It is a very stable OS.
Now, as far as the process to upgrade, need to first ask some questions:
1. Have you ever done any disk cleanup/maintenance/repairs, from a software perspective? You actually can do a good amount of disk cleanup on your own, and there are a number of excellent products available (both free and commercial) that can help you with those tasks.
2. This is
critical: are you making backups to an external device? That is a mandatory part of this process. Snow Leopard does come with Time Machine, and it is "ok" for backups. However, for this project (and any future ones), you might want to consider using either of the following programs:
SuperDuper! -
https://www.shirt-pocket.com/SuperDuper/SuperDuperDescription.html
Carbon Copy Cloner -
https://bombich.com/
Both of them make a
bootable backup/clone of your system. That's right, I said
bootable. It's almost like having another machine, and it will make this installation of Yosemite so much easier, faster, and smoother. (I use SuperDuper!, and it works like a charm).
3. What third party software (ie, non-Apple) do you have? You will most likely need to upgrade some/most/all of those programs to be compatible with Yosemite. This link is helpful with that:
https://roaringapps.com/apps
4. Does your Mac still contain the original hard drive it shipped with? And how much free space is on it?
Once you let us know the answer to those questions, we can help you better.