On the Unity Editor open your project and:
If you already created your empty git repo on-line (eg. github.com) now it's time to upload your code. Open a command prompt and follow the next steps:
cd to/your/unity/project/folder
git init
git add *
git commit -m "First commit"
git remote add origin [email protected]:username/project.git
git push -u origin master
You should now open your Unity project while holding down the Option or the Left Alt key. This will force Unity to recreate the Library directory (this step might not be necessary since I've seen Unity recreating the Library directory even if you don't hold down any key).
Finally have git ignore the Library and Temp directories so that they won’t be pushed to the server. Add them to the .gitignore file and push the ignore to the server. Remember that you'll only commit the Assets and ProjectSettings directories.
And here's my own .gitignore recipe for my Unity projects:
# =============== #
# Unity generated #
# =============== #
Temp/
Obj/
UnityGenerated/
Library/
Assets/AssetStoreTools*
# ===================================== #
# Visual Studio / MonoDevelop generated #
# ===================================== #
ExportedObj/
*.svd
*.userprefs
*.csproj
*.pidb
*.suo
*.sln
*.user
*.unityproj
*.booproj
# ============ #
# OS generated #
# ============ #
.DS_Store
.DS_Store?
._*
.Spotlight-V100
.Trashes
Icon?
ehthumbs.db
Thumbs.db