Installing Digital Audio Workstations (software in general) - Methods needing explanations

Those packages should both be in Trisquel's repository. Any package in Debian should generally be available in Ubuntu and Trisquel unless it is too new or excluded for some particular reason, such as FSDG compatibility.

Either run

$ sudo apt install rosegarden puredata

or search for and install those packages via Synaptic or Add/Remove Applications.

> I see under the release for Debian it says 'Official Debian package ("apt-get")'. When I click on that link, I am directed to a repository for many releases.

Those are all the versions in Debian's repository.

https://deb.debian.org/debian/pool/main/p/puredata/

They aren't meant to be installed manually. Debian users are meant to install them via the package manager. The equivalent files for Trisquel are here

https://archive.trisquel.info/trisquel/pool/main/p/puredata/

for puredata and here

https://archive.trisquel.info/trisquel/pool/main/r/rosegarden/

for rosegarden. Again though, you aren't meant to directly use this page yourself. You use the package manager, which will know which version to select.

Hi, jaisgossman! You posted some good questions!

Chaosmonk has already posted the relevant command for installing the programs, so I'll just add, that whenever I want to check if something is already available in my distribution, I just enter
$ apt search *program-name*

for example:
$ apt search rosegarden
Sorting... Done
Full Text Search... Done
rosegarden/stable 1:16.06-1+b1 amd64
music editor and MIDI/audio sequencer

Some names might yield more results when searched for, i.e.
$ apt search pacman
Sorting... Done
Full Text Search... Done
hannah/stable 1.0-3+b1 amd64
pacman-like game, child oriented

hannah-data/stable 1.0-3 all
pacman-like game, child oriented - data files

njam/stable 1.25-9+b1 amd64
pacman-like game with multiplayer support

njam-data/stable 1.25-9 all
pacman-like game with multiplayer support -- data files

pacapt/stable 2.3.13-1 all
Arch's pacman-like package manager for some Unices

pacman/stable 10-17.2+b1 amd64
Chase Monsters in a Labyrinth

pacman4console/stable 1.3-1+b3 amd64
ncurses-based pacman game

(Note, that I'm using some other dpkg-based distro, so package versions might not match those in Trisquel)
Of course, You can also perform the search using some gui frontend, as Chaosmonk has already noted.

In case You want to use cli, though, I encourage You to find some tutorial on apt usage (there're many on the web). And You can also just do `man apt` ;)

As to tarballs, there's no standard way. Many pieces of software offer source in .tar.* but they use different build systems. If a program uses autotools, a simple `./configure && make && sudo make install` from within the extracted directory may suffice (although you might need/want to pass some configuration options, too). Some other software uses just Makefiles, some uses CMake, etc. However, You might also stumble upon binaries distributed as tarballs. I've even seen some proprietary software available this way...
In most cases there's going to be some README file in the tarball, which should contain the instructions for installation.

As you can see, tarballs are not a very comfortable way of installing software. That's why we have package repositories and package managers :p (And also for simple updates, better security, enforcing policy on packages (e.g.removing proprietary bits), automatic dependency resolution, etc., etc...).

Some distros' (arch, slackware) package files may also have names ending with .tar.*
As we're in Debian family, those are most likely not the kind of tarballs You want...