There are many ways to get data from the GT-31 into OSM, and it can all get a bit confusing. Here is a very simple method for beginners (like me). It looks daunting, with such a large number of steps, but it only takes a few minutes. I'm assuming that you have already played with the OSM on-screen editor (Potlatch) and know how to use it.

Put a formatted SD card in the GT-31 and switch on.
Wait for the first set of beeps and the "Card Found" message.
Go the the Memory Card menu and set Log Switch=1 second, Min Speed=1 kmh.
Wait for the second set of beeps and the "New file" message. This means that the unit has started recording points.
Start your journey.
At the end of your journey, switch off the GT-31.
Remove the SD card and plug it into your computer.
Run JOSM and hit the "Open File" button.
Load the latest file from the SD card. It will be in a folder called GPS_DATA and have the extension .TXT. The track will appear in JOSM as a thin grey line.
Right-click on the name of your track file in the JOSM layers window, and select "Convert to data layer". Ignore the warning, because we're not going to upload the raw data to the map.
Delete any parts of the track that you don't want.
Right-click on your data layer again in the JOSM layers window, and save it as a GPX file on your local disk.
Close JOSM.
Open OSM in your web browser and log in.
Click the GPS Traces tab.
Upload the GPX file that you just created. A little picture of your track will appear on your GPS Traces page, marked with a red "PENDING" label.
Wait a few minutes and refresh the page. The "PENDING" label should have disappeared, meaning that your track has been added to the database.
Click the blue "Edit" link next the track name.
You are now in the OSM editor. Any existing OSM data will appear, together with your new track in light blue.
Edit OSM.
Save your changes.
You're done.

Notice that I didn't use the GT-31's Track menu. This gives access to lots of extra features, but isn't necessary when you're only collecting data for OSM.

If you don't want to use an SD card, you can log data into the GT-31's internal memory instead. Perhaps I will describe that procedure when I have had some practice at it.