Bags for longer touring
Short answer based on the little information given I'd guess two larger panniers would be enough, especially if the tent and sleeping mat is strapped on top.  Watch out for the heel clearance, especially with the largest ones.

Long answer and rambling: The length of the trip doesn't matter as much as you'd might think.  It really depends on how much you'll going to pack.  The truth is that probably a beginner will back more than an experienced bicycler tourer because you have to figure out for yourself what items and luxuries you'd like and which ones aren't of value to you.

Make a list of the things you would like to take along.  Maybe get some feedback from experienced tourers and have them review and comment on your list.  Take their suggestion with a grain of salt, since it's really personal what one takes along.  Then judge from that how large of a panniers you need.  Don't over stuff them if possible.  Not only will it be harder to get your things out of the bags but you also need some room for food purchases during your trip.  Often people stop at a grocery just before the campsite to stock up on food for the night.

Don't carry water in you panniers.  Two or three water bottles in cages should be plenty.  You wan want to take a small water bladder along if you go through the occasional area where extra water is needed (i.e. campground where drinking water isn't available).

Ortlib and Arkel are some popular pannier bags but really any sturdy pannier bags will work.  Some people even mount kitty litter tubs as pannniers to their rack.  Ortlib bags are by default rain proved but they also don't have external pockets and aren't breathable.  Most other bags including the Arkle need a rain cover for rain. BTW, try choosing a color that improves your visibility (yellow or red instead of black).

A main thing to keep in mind is to balance the weight evenly between the sides.  If carrying much you can also mount a front rack.  This will also distribute the weight off your rear wheel.  The goal with front racks is to keep the center of gravity low.  So lowrider front racks are often preferred.

Some alternatives to pannier bags are trailers (single wheeled like the bob trailer or two wheeled child trailer), frame bag (especially popular with mountain bikers in conjunction with handle bar rolls when riding narrow trails), or just tying everything onto your rack.

[links are just for clarifications, not for endorsement]