There are a number of ways to download the source code. Some are described here, an alternative all-inclusive guide is also located at https://wiki.boomerstation.space/Downloading_the_source_code
Follow this: https://wiki.boomerstation.space/Setting_up_git
Download the source code as a zip by clicking the ZIP button in the code tab of https://github.com/tgstation/tgstation (note: this will use a lot of bandwidth if you wish to update and is a lot of hassle if you want to make any changes at all, so it's not recommended.)
WARNING: OPTIONS 3 AND 4 ARE NOT AVAILABLE FOR WASPCODE, THESE ARE FOR /TG/ ONLY AS OF NOW
Download a pre-compiled nightly at https://tgstation13.download/nightlies/ (same caveats as option 2)
Warning: option 4 is out of date, and not maintained, use at your own risk
Use our docker image that tracks the master branch (See commits for build status. Again, same caveats as option 2)
docker run -d -p <your port>:1337 -v /path/to/your/config:/tgstation/config -v /path/to/your/data:/tgstation/data tgstation/tgstation <dream daemon options i.e. -public or -params>