How do I add more circuit breakers once the breaker box is full?
The only reason to replace the entire box is if aesthetics: if there isn't enough room on the wall for two boxes, or the box is in a closet or cabinet where there isn't physical space.

Putting in a new breaker box is much easier than replacing an entire box.  If you replace the entire box, you will need to rewire every breaker.  When the original box was wired there was lots of slack on each wire, but after each breaker was installed the wires would have been trimmed.  A good electrician will allow slack in the wires for reconfiguring breakers, etc. but if it's a small box you may not have enough wire, or you may end up running the wires in ways that aren't as neat or professional as it should be.

With a new box, on the other hand, you only need to remove one 220V or two adjacent 110V breakers.  The old wires can be run into a junction box to provide as much length as you need for wiring into the new box. 

As Mike Sherov noted, you need to be sure that your service can handle the load and number of circuits.  That'll be a location specific code issue.  If the knob-and-tube wiring was recently replaced I'd be surprised if the old service (probably 50 or 60 amp) wasn't upgraded as part of that process (likely to 150 or 200 amps). 

If your new circuits are within the capacity of your service and local regulations permit, this is a job you can do yourself.  If you choose to do it yourself, in some jurisdictions you can ask the inspector if your plan will keep you in code, to save the cost of a consultation with an electrician.  (In some places of course, they're too busy for such questions.)  Or you can research it yourself: it's not difficult if you have access to a code book.