I will assume you intend to develop on Windows.
The easiest & quickest way to start is to use a pre-compiled/pre-installed psl1ght sdk like the PS3SDKv2.zip archive that is widely available online. Alternatively, you can download this repository as a zip at
https://github.com/Joonie86/PSDK3v2, the resulting zip should be the same.
It's an older setup based on gcc 4.x which already includes mingw/msys, you just unzip the archive at the root of any drive on your PC, inside the newly extracted PS3SDKv2 folder unzip the 7z archives & the sdk is ready to compile your code (remember to set up system environment variables globally or you will need to take care of setting them up within batch files for each project).
I recommend using this sdk version if you are new to all this & wish to learn because all existing homebrew sources for psl1ght should compile with it without problems.
However
if you are already comfortable with sdk deployments in general, can cope with any installation issue that may arise & like to experiment and optimise, you can use the latest sources from github (based on gcc 7.x) & compile/install the sdk using available scripts. All necessary files to compile/install the toolchain & additional libraries can be found in repos at:
https://github.com/ps3dev/
https://github.com/bucanero/
Keep in mind that nearly all available psl1ght based homebrews were developed on gcc 4.x, some homebrew may not compile or may malfunction when compiled with gcc 7.x, you may have to tweak the source code.
There is documentation available for psl1ght sdk, no doubt you will need it so I suggest that you get familiar with it quickly.
https://zerkman.sector1.fr/ps3/psl1ght/
If you like to work with an IDE on Windows, there is a Visual Studio plugin for psl1ght sdk.
https://github.com/Dnawrkshp/PS3-VS-Integration
FYI
The only alternative to open source psl1ght sdk is the official sdk by s#ny, unfortunately it is still under license.
The official sdk is the only sdk that can compile prx modules (more or less the equivalent of DLL files in Windows or SO files in Linux), keep that in mind if you plan to develop a sprx project. Psl1ght sdk can only compile elf projects.
The official sdk contains a lot of information, the documentation & samples are quite extensive. Various official versions have been leaked, the latest one being a full leak of the 4.75 sdk.
Before you ask, we cannot help you to get a copy of the official sdk, unless you own a license, you are on your own there.
Without DEX & the debugging tools available in the official sdk (or alternatively the IDA debugger), you won't be able to debug your code easily though, no breakpoints, no stepping, no console output, no register/stack peeking etc...
So I recommend that you install the "debug" HEN payload rather than the "release" payload on your console. This way you will be able to receive HEN broadcasted debug data on UDP with socat (port 18194), that includes stuff like printf calls which may prove useful to test your code output at various stages.
With the sdk installed, you should also be able to recompile the HEN debug payload yourself & if required you could add your own kernel hooks and monitor any data you wish to debug, but I guess this is advanced stuff to get into at a later stage.
