If I move A Nether Portal and relight it, will it still work?
If by work you mean connect to the same one in the Nether, it should if you don't move it too far. The rules governing Nether Gates are far more complicated than I care for, but if I remember correctly, the game connects to a portal within 128 block radius topside. This means that if you don't move it very far (128 blocks) it should still connect without creating a new portal. If you mean will it relight, yes it will. You can always check the wiki for more details (http://www.minecraftwiki.net/wiki/Minecraft_Wiki), or there is a topic on Nether Gate linking in the survival section somewhere.
Can I ask what exactly is Lapus Lazuli and how can you get to The End?
Lapis Lazuli is a mineral that is often confused with diamond by newbies. Breaking the block with a pickaxe yields lapis lazuli dye which can be used to color wool. Going to The End requires that you find a Stronghold and repair the End Portal located there. This is the wiki article: http://www.minecraftwiki.net/wiki/Ender_Portal