Like others wrote the basic languages to configure FPGA's are verilog and vhdl. I myself choose verilog because it resembles C, but the pitfall is sequential thinking used when programming in C, will give you problems when working with FPGA's. The logic here is far more parallel then sequential.
It's more than just parallel vs sequential. HDL is not a programming language in a conventional sense, it only looks like one. When you write a program you are writing a series of instructions that run in series. When you write HDL, you are describing a digital logic circuit in a language that superficially resembles a program, but in reality it is closer to describing a schematic.
My first experience with FPGA's was based on drawing schematics. Old Xilinx XC3000 series with XACT for translating, placing and routing and Orcad for drawing the schematics. Used Xilinx libraries with all kinds of logic parts. Much like what now is called vendor IP. Being used to schematics I found it very easy to do. Writing it up in a language like verilog or vhdl makes it harder to see the intent, at least for me, having little experience with it. That is what you get when you leave the hardware game for software. Only returned to playing with MCU's and FPGA's a couple of years back, and done one actual project with a FPGA creating my own AWG and reverse engineered the configuration of another
It takes its time to learn, but, in my opinion, having basic digital hardware knowledge is useful, both for doing software and (FPGA) hardware. The OP wrote that he has some experience cooking up his own hardware so guess that part is fulfilled.
Coming from a software background means having to loose the sequential and programming way of thinking. That is why I wrote "configure" instead of programming. I know it is describing hardware, but the languages do bring constructs of both worlds.
The OP asks for how others got started, well as I wrote above, a long time ago just dove in with the need for making smaller hardware with more functionality, the foundation I worked for sent me on a short course about the Xilinx parts and software, and I designed my first systems with it. Did some work with them for another company and then got lost in the software world where there was more work to be found. Now retired I returned to the game and bought a couple of boards to play with.
Did the AWG project after reading up on verilog on the different websites and had some help from forum member BrianHG who gave excellent advice. The result of that project is here.
https://github.com/pecostm32/Lichee_Nano/tree/main/FPGA/signal_generator Still have to finish the software for it, but with to many hobbies, these thing can get shelved for a while