The first thing that you need is a good development board that won't break your bank account. FPGA hardware is decently expensive, which makes the boards pricey. It's not the same with other types of hardware like microcontrollers where you can get many boards for pretty cheap.
With that in mind, a great dev board for people starting out is the Arty A7 by Digilent. For around $130 bucks, you can get a decent FPGA and right out of the box play around with switches, buttons, and LEDs. On top of that, it has DDR3 and Ethernet so that you can move on to more advanced interfaces.
The board can be found here: https://store.digilentinc.com/arty-a7-artix-7-fpga-development-board-for-makers-and-hobbyists/
Then, with the PMOD connectors, it opens up a whole new world of add on functionality. Check out the selection here: https://store.digilentinc.com/pmod-modules-connectors/
For example, you can get LED screens, bluetooth adapters, GPS, accelerometers, digital to analog converters, and analog to digital converters to name a few.
With that in mind, you can come up with all sorts of projects from the different interfaces and capabilities. Here are some ideas to start with the above hardware:
1. Switch detector to LED status
2. Push button debounce circuit
3. Video test pattern generator and DDR3 frame memory driving a VGA display (PMODs: VGA)
4. Ethernet interface
5. Handheld board controller that steers a block on a VGA screen (PMODs: VGA, Accelerometer)
6. Mini oscilloscope for low frequency (PMODs: A/D, VGA)
7. GPS logger (PMODs: GPS, microSD card)
8. Temperature logger (PMODs: Temperature sensor, microSD card)
9. Analog digitizer and analog regenerator (PMODs: A/D, D/A)
10. RS232 serial port interface (PMODs: RS232)