The Minecrafts worlds are backwards compatible.
You will not have any problems. Minecraft will automatically upgrade them.
I played very very long with an Alpha world, very long. Never had any problems besides that I have no biomes, no pumpkins or anything else that was added later.
Please consider that the worlds are backwards compatible but only partly upwards compatible. This means that your game will crash if you load a game with new biomes, blocks or items inside of an old version which doesn't know about these items.
You can safely upgrade your Minecraft. Your Savegames will stay compatible. You can create backups if you are really worried but I never had problems with Mojangs Algorithm.
Have a nice day