Build state-machines or cpu using logic gates in logisim. It sounds complex but there are only a handful of primitives you have to understand. Learning that FPGA conditionals are equivalent to multiplexers is much easier to learn when you make one by hand.

Depends on which peripherals you have.

* A good start is a stop watch (if you have a 7seg and buttons available)

* Implement a digital function generator without using look-up tables. (Can read values over serial)

* If you have access to an ADC and DAC implement an echo filter, then a tuneable butterworth

Start simple. Implement an adder, configured by some switches. Represent the output in binary on some LED's.

This will help you wrap your head around some of the important basics, and then you can go from there.

It is very difficult to debug FPGAs so it is important to "build up" to what you want in piece by easy to verify piece, instead of trying to build the whole thing at once and debug it later.

http://www.fpga4fun.com has some nice projects for starting out with FPGAs.

find some other chip you like, and write an interface for it.