Depth of cut tends to be one of the most important aspects to using a router that is often overlooked.

Generally speaking, you're better off taking very thin cuts when using a router. This allows for consistent feed speeds without bogging down the router bit speed.

If you're working with woods of varying density (ie. knots in the wood) then you'll want to size your cut depth based upon the worst case of what you expect to see.

The balancing act is taking lots of thin cuts takes more time and we often want to move on to the next portion of the project. So sometimes you cut a little deeper and take the risk of scorching, knowing that you'll have to sand that out later.

To address some of your questions:

Should you see better results using a routing table instead of locking the board and using the router freehand?

A router table can provide better support for the material so you'll have more consistent results. But some pieces won't fit on a router table and you need to go freehand instead.

• Old vs. new router bits

Age of the bit isn't as much of a question as sharpness of the bit. Given that older bits tend to be more used and therefore less sharp, newer, sharper bits are the better way to go.

• Speed of the router (or power output), does it make a difference?

You need to adjust the speed of the router based upon the size of the cut you're making. Big bits need to be run at lower speeds. Smaller bits can be run at higher speeds. The primary concern is the friction generated between the bit surface and the wood.
Big bits run at high speeds generate more heat from friction which leads to a greater chance of scorching the wood.

The direction of the cut makes a difference. Normally we cut against the spin of the bit, clockwise on the inside of a frame and counterclockwise on the outside. If you are concerned about burns, you can go with the spin of the bit(a climb cut) rather than against it. It's even more important to make shallow cuts with this, because a deep cut can make the router try to run off from you, or "climb out" of the proper position (hence the name).

A safety note regarding climb cuts: This is for hand-held routers only. Do not perform climb cuts on a router table as it may yank the workpiece away and pull your hands into the spinning bit. With a hand-held router the worst that's likely to happen is you have a little scalloping to remove

Other than that, sharp, clean blades, a lower spindle speed, and shallow passes are all helpful in preventing burns. Keep your feed rate steady, lingering will cause burns as well.

And because the internet loves list, here's
Rockler's list of ways to ensure a good cut:

1. Use bits with 1/2" shanks for better stability
2. Ensure the bit is properly installed (all the way in then back out 1/8")
3. Ensure bits are sharp, smooth, and clean. Ensure guide bearings spin freely.
4. Control your bit speed. Larger bits need slower speeds.
5. Connect a shop vac to the router for chip removal
6. Make multiple light passes rather than one heavy cut
7. Too slow will burn the wood, too fast will make a rough cut
8. Keep the workpiece from moving. Clamps are good. Rockler seems to think their "bench Cookies" are good. I've never used them.
9. Route in the correct direction. They give the "traditional" non-climbing cut (counter-clockwise on the outside, clockwise on the inside) but see above regarding using a climbing cut to avoid burns.
10. Rout edges in the right order, end-grain first then long-grain, this will remove tear-out from the end-grain edges being routed.

An oft overlooked reason for burning is excessive pitch build up on the bits. Use a bit cleaner (I use Simple Green) and nylon or brass brush to clean the bit and it will cut much cleaner.

The type of wood can make a difference.

For instance cherry is a wood that shows burn marks more readily than other hardwoods and is great for testing other means of preventing charring mentioned in other answers, i.e. depth of cut, direction of cut, dull or dirty bits.