I can recommend a tent and sleeping bag. I'm not going to recommend a backpack as you really don't want oneat least, not if you mean to carry this gear while bicycling. You want the weight low and on the bike, not on your back. You have two options, depending on your bike design. If your bike has no rear suspension, then you should add a luggage rack (Jandd Expedition Rack: $57; www.jandd.com) and a couple of rear panniers (Performance TransIt Epic Panniers: $100 per pair, 2,800-cubic-inch capacity; www.performancebike.com). The panniers give you enough room to stash your clothes, stove, and food; the top of the rack carries the tent and sleeping bag, secured by bungee cords or other straps.
If your bike is a softtail, then you'll need a trailer. Get a single-wheel model so it tracks on narrow paths. The classic here is the BOB Yak Plus, a well-made trailer that comes with a big, waterproof bag for $289 (www.bobtrailers.com). Trailers are an excellent way to lug gear on a bikethey take the weight off the frame and are more aerodynamic. They do also make for some odd handling characteristics, but then so do panniers.
As for a tent, REI's new Roadster ($129; www.rei.com) is an excellent one-person tent that weighs a mere two pounds, ten ounces. It even has a vestibule so you can keep shoes and stuff out of the rain. Another good option is the Marmot AT ($179, three pounds, eight ounces; www.marmot.com), a slightly larger tent that uses single-wall construction to cut down the weight.
There are lots of choices in sleeping bags. Get a bag rated 30 to 40 degrees, down or synthetic depending on your price sensitivity. Mountain Hardwear's new Phantom ($240, one pound, five ounces; www.mountainhardwear.com) is a 35-degree down bag that would fit the bill. In synthetics, take a look at Integral Designs' Primaloft-filled, 40-degree-rated Andromeda Strain ($170; www.integraldesigns.com).