TFVC
TFVC is notoriously hard to migrate from one server to the other. There are some tools out there that can help, but TFVC->TFVC is hard to do right. Due to the fact that you can do things in TFVC that aren't possible in Git, it is not always possible to carry over **all* history and **all* branches. In general, though, it's enough to transfer the main branch with some history.
With that caveat, fortunately, TFVC->Git is a lot easier. From your current Azure DevOps account, choose the Import Repository option:

You can specify a single branch to import with up to 180 days of history. For most projects that's more than enough. Using this route it's not possible to import multiple branches, so you'll need to integrate your work into a single branch first in case you have in-progress items.

This will convert your TFVC branch into a Git repository. You can execute this as many times as you want to import different branches and from different TFVC repositories until all your work is in Git.
Git
There are multiple ways to move a Git repository from one system to another, but since you're using Azure DevOps, you can use the same Import Repository option to let Azure Repos import the data from the old account straight into a new one.
First, you'll need to create a new target account (or use your existing personal account if you have one). Then use the same Import Repository option, but this time from your new account.

Enter the Clone URL for the repository you want to move over and provide your Git Credentials or a Personal Access Token:

This will automatically pull in all your code, branches, tags etc into your target account and into a new Git repository. Execute this sequence as many times as you have Git repos.
The command line way
You can also perform the same steps using the commandline:
Git
To clone a repo for migration use a fresh clone and specify the --mirror
option:
git clone https://dev.azure.com/{Org}/{Project}/_git/{Repo} --mirror
Then push everything into a new, empty, git repository on the new account:
git push --mirror https://dev.azure.com/{NewOrg}/{NewProject}/_git/{NewRepo}
TFVC
For TFVC you have tools that can import a TFVC branch into a git repository with optional history as well. git tfs
allows you to perform such an export:
git tfs clone http://your-tfs-server:8080/tfs/your-collection $/your-tfvc-repo
This process will take some time to import your changesets into the local git repo. Afterwards, you can push the resulting Git Repository into the new target Azure DevOps project:
git remote add target https://dev.azure.com/{NewOrg}/{NewProject}/_git/{NewRepo}
git push --all target