The iD editor built into the openstreetmap.org site is really easy to use. Sign up. Navigate to where your trails are. At the top left of the page, hit 'Edit'. At the top of the map area, select the Add Feature > 'Line'. Left click on the map to start drawing your line. Press escape on your keyboard to quit drawing. You will be prompted to select a feature type, which you'll probably want a Path or Foot Path (click the 'i' icon next to the feature for more information from the wiki which is very helpful).

Keep in mind no changes are made until you commit them using the 'Save' button at the top right of the map area. You also have 'Undo' available. You can also make changes to your line before saving. Right clicking the line or one of the nodes on the line will give additional options (such as being able to continue drawing your line).

Hit 'Save', add a descriptive Changeset Comment and add your source (usually aerial imagery).

If the trail is tree-covered and not visible on the map, you may have used GPS to record the track. In that case you would want to use something like the JOSM desktop editor that has more capability. It can import GPX files, etc.