Hi,
We have done our best to make this process as simple as possible. In the vast majority of cases, all you need to do is download the newer version of the Engine. Run the Editor to open the Project browser and find your project. It should look like it is grayed out, and may indicate the prior Engine version in the corner of the image for the project. Select the project and click the Open button (or double-click on the project). You will see a window appear asking how you want to convert the project. The recommendation is to always select the Open a copy option, as this will make a copy of the existing project and upgrade it to the new Engine version. That way if something goes wrong, the original project is still there for you to go back to
You may need to make some changes to the project at this point to get everything working in the new version of the Engine. If you have ever looked through the list of changes, fixes, and new features that accompany the release of a new Engine version, you will know that there are frequently extensive changes that could possibly affect your project. Ideally, there should be no new problems appearing, but as with any large piece of software there is a possibility that one of the updates has broken something for your project. Please let us know if this is the case.
Once you have verified that your project is working in the new version of the Engine, you can delete the original project. Also, keep in mind that it is often a good idea to upgrade your project one Engine version at a time. If your project is still using 4.8, don’t go straight to 4.10. Upgrade to 4.9 first, make sure everything is working, then upgrade to 4.10.