So you are stuck with SVN, huh? So here is what you will do:

git svn clone -s --prefix=origin/ https://svn.url/projects/{my_project}

This assumes that you have {my_project}/trunk, {my_project}/tags and {my_project}/branches

Work the same way as you work with git, except that:

Instead of pushing you:

git svn dcommit

And instead of pulling you:

git svn rebase

You can fetch by running the command:

git svn fetch

Avoid pushing/pulling to/form different sources. Use it as it was not a distributed versioning system.

Here is an example of script for testing how you can use git-svn.