I always have trouble with flush trimming. I use clamps to keep the piece in place but it gets in the way.
Is it better to use a table router or a hand router?
Couldn't I use some sort of double sided tape?
How much should I remove of the wood before flush trimming the other thing to it?
Table vs. handheld depends on what you are trimming flush to what and whether it's easier to move the workpiece or the tool.
"Carpet tape" is the woodworking standard double-sided tape.
For cleanest results, minimize the amount being removed by the router; if there is too much you may have to take several progressive passes. Trimming to about 1/8" short of the final shape is plenty close enough and retains some space for cutting error.
Is it better to use a table router or a hand router?
If you ask three router users this question it's likely you'll get four answers to this. Obviously doing it either way can work and give good results, because there are people who do it each way. And as keshlam's Answer also covers the size of the workpiece is a factor.
If the piece is small it really calls for the router to be fixed and the workpiece to move (using push sticks to protect fingers as appropriate).
If the piece is large and provides a good registration surface for the router baseplate then it's fairly easy to get good results holding the router so it can make sense to do it that way.
Couldn't I use some sort of double sided tape?
You could, and some people have great success with it. But there's a better way: using masking tape on both pieces and applying superglue in between, see previous Answer here: How do I temporarily attach two pieces of wood together for machining?
This has a few key advantages, no. 1 of which is zero slip — it's essentially impossible to get the pieces to slip. And there's nothing worse than working two pieces that are held together with double-sided and having them go out of register! Even 1/64" (0.4mm) is enough to be a problem and that is well within the 'wiggle room' for some double-sided tapes.
How much should I remove of the wood before flush trimming the other thing to it?
There's no one answer to this but perhaps as much as possible is the thing to keep in mind. Anything you can do to reduce the load on the router and its bit is good practice, the ubiquitous advice to prefer multiple light/shallow passes rather than doing one heavy cut is the obvious face of this advice.
Light passes can greatly help to improve the routed surface (skimming cuts in particular can leave a zero-tearout, finish-ready surface with a sharp bit and the right feed rate), as well helping to reduce wear on the bit and the router's bearings.
To put some numbers on this, aim to cut to under 1/8" from your finished line, and 1/16" if you can.
