How to stop the lagging on Minecraft?

I'm having an issue on Minecraft. Ever since it came out, I've been using version 1.7.10. I have a very good computer, and it runs well. I don't run Minecraft on Java anymore, since they came out with an alternative. I play with most of the graphics on the highest settings.
But when I decided to change to the latest version (1.8.8) it had a TON of lag spikes, and it's absolutely unplayable! On Single Player and Multiplayer, with the graphics on the lowest settings. I want the new fun items (rabbits, guardians, etc.), and I really want to be able to play Multiplayer again.
My computer is free of viruses, it runs very well with other games, etc., but what is there I can do? I don't want to have to stay on version 1.7.10 forever. Please tell me what I can do.

Any help is appreciated.

If you've already tried adjusting all of the graphics settings without success, you should try just reinstalling the game.