Beginner box?
Buy things like D0 and D1
IF you're going into APs that's more along the lines of the full ruleset. I suppose you can, but you might not have everything you need to DM more complex NPCs because you might not have the rules.
I just bought the beginner box and would like to continue a adventure after we finish it what adventure should I get to follow up after it. I was thinking Rise of the Runelords but if not im open to suggestions.
Check out the transitions pdf, free download:
http://paizo.com/pathfinder/pathfinderRPG/products/beginnerbox
Will help you get going from beginner box to the 'main' ruleset.
I agree with the other poster, try some of the great modules available. There are also tons of PF compatible products out there. If you're ready, you can also start to create your own adventures for your players.
I'd recommend picking up the Core Rulebook and the GM Guide. READ the GM Guide - it's a fantastic resource and very well put together. I think it should be standard reading for players as well, as it spills tons of great advice on how to work together in a party. Many great ideas on how to create dungeons, traps, encounters, and run your campaigns.
Rise of the Runelords would be lots of fun, but if you and your players are just starting out I'd highly recommend a few modules and home brew games. It will also give them the opportunity to switch it up with new characters. You can allow them to make characters starting at higher levels, if you wanted to run a 5th level module let's say. Nothing wrong with that. Just have them work together to build some back story as to how they came together and why they are a party. That can go a long way in a game. Something like the PF Adventure Paths can take 6 months to a year to complete with regular game sessions. So it may be lots more realistic and fun to make your own adventures or run a fun premade module. Good luck and most of all have fun!
If you want you can also make your own world and adventure to follow. You can segue from some aspect, place, person, item from your first adventure and continue it into your next to start your new adventure. This side adventure will now be your main adventure for a while and from that wil have plot hooks, side stories, branched adventures, quest adventures, and objective adventures.
Example of possible segues...
1. One of the people one of your players spoke to happens to pay more attention to your collective more than usual. Make this person either a potential bad guy later on, wo wish I'll omens on the group for foiling one or more of his plans, if a big boss it'll be multiple. Or it could be a good natured person who wishes to follow the deeds of your group, possibly publishing stories, or write about them for the public as news, could be good or bad for your grop. Or even have the person needing our groups help to locate something or someone for them, and since they already somewhat know this character it won't be too far off to help them now.
2. Have a particular item one of the players character wants/ needs for a quest. Have the item be something barely mentioned in the first adventure, but now is more prevelant due to being stolen or misplaced. Or have the item be from a character, or even a players backstory and is relevant to why they're in the group and why they are adventuring.
......
If none of these work for you, then as mentioned some of the earlier adventure paths work great. If you then do decide to branch out and do things yourself, these examples will still come in handy.