In short: Best practice is branch out, merge often and keep always in sync.

There are pretty clear conventions about keeping your code in a separate branches from master branch:

1. You are about to make an implementation of major or disruptive change
2. You are about to make some changes that might not be used
3. You want to experiment on something that you are not sure it will work
4. When you are told to branch out, others might have something they need to do in master

Rule of thumb is after branching out, you should keep in sync with the master branch. Because eventually you need to merge it back to master. In order to avoid a huge complicated mess of conflicts when merging back, you should commit often, merge often.

Good practices to follow

A successful Git branching model by Vincent Driessen has good suggestions. If this branching model appeals to you consider the flow extension to git. Others have commented about flow.

Tagging practices

As you already know, Git gives you commit identifiers like 1.0-2-g1ab3183 but those are not tags! Tagging is done with git tag, and the tags that are created using git tag are the base for the commit identifiers git describe creates. In another words, in Git you don't tag branches. You are tagging commits. It is correct to say that tag is just an annotated pointer to a commit.

Lets look at practical example that demonstrated it,

                        /-- [v1.0]
                       v
---.---.---.---S---.---A     <-- master
                         \ 
                           \-.---B     <-- test

Let's commit 'S' be commit pointed by tag 'v1.0'. This commit is both on branch 'master' and on branch 'test'. If you run "git describe" on top of commit 'A' (top of 'master' branch) you would get something like v1.0-2-g9c116e9. If you run "git describe" on top of commit 'B' ( aka the 'test' branch) you would get something like v1.0-2-g3f55e41 , that is the case with default git-describe configuration. Note that this result is slightly different. v1.0-2-g9c116e9 means that we are at commit with sortened SHA-1 id of 9c116e9, 2 commits after tag v1.0. There is no tag v1.0-2!

If you want your tag to appear only on branch 'master', you can create new commit (e.g. only update default / fallback version information in GIT-VERSION-FILE) after branching point of 'test' branch. If you tag commits on 'test' branch with e.g. 'v1.0.3` it would be visible only from 'test'.

References

I have found many, many, useful blogs and posts to learn from. However, the ones that are professionally illustrated are rare ones. Thus, I would like to recommend a post - A successful Git branching model by @nvie. I have borrowed his illustration :)

First of all, since your specifications are rather unclear, I'm not sure how much of this really applies. But I'll give it a shot...

If your project has one single release cycle, you only need one repository. If your three teams need to release new version independently, it gets a lot messier. But for that I'd need more specific input. At the moment, it sounds to me like you need to stabilize your API, develop the library and manage your back-end development. If that's part of one project, go ahead and use one repository.

As for the branching, I'd recommend to always have one branch that simply represents the latest released and working version. This branch will probably mostly have merge commits. I like the idea of release branches independent of development branches as a release manager could then easily review everything (after feature freeze if you want so), scratch the last few itches, set the version properly, and merge into your production branch. (You could even bind hooks or deployment strategies to that merge in order to automatically push release onto your build system or something).

As I said above, I'd recommend having one release cycle within one repository. If you can have that, only one production branch will be necessary. If the teams really need to release new versions of the same project independently, it feels like there's something wrong with the project management. But that's again without knowing much about the team and project in general.

A style that I like much is the git flow as described by Atlassian. It suggests having one master (production) and one development branch. On development the actual work happens but, that usually splits up into feature branches. What I like is that a development manager (someone with deep technical insight) can manage the development branch. They decide whether to merge new features or not. At some point they will tell a project/release manager that the team's done and that project/release manager can then (together with the development manager) work out the last few glitches on a release branch before it'll be merged into master (and back into development, of course). It's just an interesting separation of concerns that developers are so keen on regarding different aspects of the very same projects. :)

Hope this helps a bit.