Does anyone have any tips for flying with a keyboard and mouse?.
My last space sim was Freelancer (which I loved) and I'm finding Freespace2 a bit more challenging.

This scheme works reasonably well for me.
W - Accelerate
S - Decelerate
A - Turn Left
D - Turn Right
Z - Roll Left
C - Roll Right
V - Comms
Shift-S - Next Subsystem
Middle Mouse Button - Countermeasure

Some commands are lost. You can reassign them or leave them blank.
Make use of Insert/Home/Page up, Down/ etc. when appropriate.
Learn what keys do what and you should be fine.

No tips needed. Simply start playing it with KB+M right from the training missions and soon you'll get accustomed to the KB+M controls. Practice, practice and more practice, it is always better than theory.
You can change the difficulty any time, by the way.

P.S. Actually, there is no need to reassign the control buttons. I'd recommend to use the default scheme because it's pretty well optimised already. And turning and pitching should be done with a mouse.

I also found the constant swiping of the mouse to be a pain - that's why I prefer my joystick :P. Rolling has some, though limited, benefit. I think most if not all ships are symmetrical in their ability to pitch vs yaw. However rolling can change the orientation of your ship relative to an attacker (depending on where they are coming from) and thus change which shield facing they're hitting. It can also change the orientation of your weapons to someone you are attacking and depending on their profile and the arrangement of your weapons you'll stand a better chance of striking their ship with more of your shots. As you practice more with a certain ship you'll find you're doing rolling instinctively to keep your target in a better firing position. You can also use it to pull of more complex maneuvers since you can change what the cardinal directions are as you roll - so for a joystick you can keep your stick in one position, but roll and create a harder target to hit while maintaining the same general direction of movement (essentially a barrel roll). For a keyboard, you can hold down a pitch/yaw button while rolling to achieve the same effect.

I find the mouse controls pretty ungainly in Freespace because you have to keep swiping it across the pad like crazy_dave said. In other games like Freelancer or Klingon Academy you click right mouse then move your mouse a certain distance and then hold it there, and your ship will move until you let go of the mouse button. I use my 360 controller for this reason.

It looks like practice is the key. Using the keyboard and mouse for navigation seems to help but I'm still pretty bad at it but getting better. (I'm not helped by the fact thet my keyboard has a different layout to the one on the reference card.)

Freespace (2) is a great game but it is giving me a hankering to dust off freelancer again :-)

Glad to hear it's (slowly) working out for you. :) You can also rebind the keys if you wish in the options menu.

When flying ships with a specific geometry, rolling can also be used to minimise your exposed surface. For instance, if you're flying a very flat ship (the Kentauroi from Blue Planet 2 is a perfect example) and are pretty close to a capital ship, you might want to have the capital ship on your left or right, instead of above or under you. A thin line is harder to hit with flak than a large polygon.

I reccomend also to use some (maybe free) mouse remapper utility (I use X-Mouse for example). This let you bind the additional buttons your mouse may have, keeping the most used stuff in your right hand.