After it has been asked a few times, I thought I'd just make a guide instead of typing the same thing over and over. Biomes are basically landscape types, e.g. jungle, desert, ocean, river, forest, plains, nether, the end, etc. You can technically get this achievement in the same world just by wandering around. You will also need to build a nether portal for another achievement. Enter the portal to add the nether biome to your list.
I found this guide very useful, which was written for the Windows 10 version of the game. Basically use these seeds while creating a new world (make sure that it is a survival world).
Just walk around in each world for a bit to make sure that you have been in (a) new biome(s). Solution for Adventuring Time in Minecraft (Win 10)
(I take no credit for this solution btw)
If you don't have the achievement by the time that you've used all of those seeds (which is unlikely), just create a random new world and run around to discover more biomes. There are more than 17 biomes, so in fact it is quite easy to obtain this achievement.