Check the Speed
Before pitting your power against the river's, it's a good idea to gather some facts about the stretch you'll be paddling. Check the river gauge nearest to where you will paddle to determine what speed the river is flowing. While your paddling speed depends on your strength, endurance and size of your craft, most people paddle at a maximum of 5 or 6 mph. That's not to say you won't make it upstream if the river is flowing faster than you can paddle. You'll just have to be clever about it.
Read the River
If the river gauge says the water flow is 5 mph, you'll find areas of water in the river that are moving much slower than this. Stay along the shoreline, especially on the inside of the turns to find the slowest current. Don't paddle under low hanging trees or near pileups of brush where the current could trap you. Wide stretches of river move slower than narrow stretches, where the narrowing of the channel creates higher pressure as the water squeezes through the tight space. Look at the surface of the water. If the water forms a "V" shape pointing downstream, it is an indication of a deep channel with fast-moving water in the middle and slower water along the banks. If the surface water flows like a "V" pointing upstream, it indicates a rock or some other obstruction below the surface of the water that is forcing the water to flow around it. Stay clear.
Paddle Into the Current
To make upstream progress, pick the slowest area of current, and paddle directly into it. This keeps the narrowest surface of your canoe or kayak against the flow of the water, offering the least resistance. Look for "Vs" pointing upstream that let you know the water is backing up against a rock. The backlogged current is known as an eddy, and provides a place you can pull into for a rest. You'll also find eddies along the shoreline behind large rocks.
Canoe vs. Kayak
While working with the river's currents is pretty much the same whether you're paddling a canoe or a kayak, there are differences between the two. A kayak paddle is heavier and longer than a canoe paddle, which can translate to accidentally clubbing rocks or tree branches when you're paddling close to shore. In a canoe, you can bring a pole to push yourself through shallows. Kayaks have the advantage of being easily turned using your body and paddle should you find the current broadsiding you. Canoes can be cumbersome and get easily swamped when turned sideways to the current.