OP:
You didn't tell us which version of the OS that you're using right now.
If it's 10.6.8, that's a good one (though old).
If Safari no longer works well enough, consider an alternative browser such as Firefox.
You don't have to go through the rigamorale for Sierra (as described in post above). You don't need Sierra.
You -can- go as far as El Capitan (10.11), but a word of caution:
Be aware that on an older Mac, with a platter-based hard drive, the more recent versions of the OS will "run", but from YOUR perspective as the user, it may seem like the OS is "walking" instead of "running". As in, somewhat slow.
Even though slow, everything should still work ok.