Go to the โExport your dataโ page and request a copy of your data. The ZIP file you will get via e-mail contains every data point Garmin has saved about you. More details about the data format can be found in the blog post โRunning on my ownโ.
You can also use tapiriik (for free) to sync GPS files between a number of destinations, including a Dropbox folder.
This will not sync manually created activities.
Alternatively, on the Garmin Connect page under โAnalyzeโ -> โActivitiesโ, you can select all activities and click on โExport to CSVโ (in the bottom left corner). This only exports the current page of activities, not all activities.