You'd need to learn some digital design for any FPGA board. I recommend the book "Digital Design and Computer Architecture" by David and Sarah Harris.
1) vivado webpack edition (ie free) lets you write (and flash) a bitstream for some of the small chips. i know it at least works for the artix-7 family because i'm doing it every day lately
2) for the artix-7 (and some lattice chips) you supposedly can use OSS (https://github.com/SymbiFlow/prjxray). i haven't tried it yet but one problem i can foresee is that the OSS tools won't infer stuff like brams and dsp. in fact the symbiflow people (i think?) explicitly call this out as the part of the project that's a work in progress.
some useful links:
https://arxiv.org/abs/1903.10407
Icestorm http://www.clifford.at/icestorm/
Yosys http://www.clifford.at/yosys/
NextPNR https://github.com/YosysHQ/nextpnr
SymbiFlow https://symbiflow.github.io/
For synthesis, the only FPGAs families currently supported by an open source flow are the Lattice iCE40 and ECP5 [0]. The latter is something you can be decently productive with and can fit quite a bit of logic (think: Amiga reimplementation, PCIe interfacing, etc.).
If you'd like to port synthesizeable code from the open source world to the commercial world, this _should_ just work as long as you're willing to rewrite any physical interfacing code (since those depend on hardware blocks available in a particular family) and stick to high-quality Verilog. But that's the same as porting across any other FPGA families.
Disclaimer: I work with SymbioticEDA, who develop and provide commercial support for some open source digital logic tooling, like Yosys and Nextpnr.
[1] - http://iverilog.icarus.com/ [2] - http://www.clifford.at/yosys/ [3] - https://github.com/YosysHQ/nextpnr