I'm trying to move the VM's data files to another datastore. I've read a lot of documentation and many of them mention right clicking and choosing "migrate". This option is unavailable to me.
Can I just use the datastore browser to move the files to another datastore, then re-add the virtual machine? Will this cause any issues?
The "migrate" option is only available with vCenter Server managing the host(s). In a single host deployment you may - as you already mentioned - use e.g. the datastore browser to move the files. If your virtual machine's files (virtual disks) are located in different folders, you may need to adjust the VM's configuration file (.vmx) though.
As mentioned there if you are running in a single host environment (and therefore probably most likely not using vCenter) you will not be able to clone a copy of the vm while it is running. If you are using vCenter you can by right clicking the vm and choosing "clone" otherwise you will be able to take copy when the vm is powered off (eg as per above post by browsing the data store (right click the data store on the main dash board screen for the host, and right-clicking the vm folder in question and copying, pls note may have more than one folder if multiple disks)