I don't use Strava but I do believe the ability to import routes is there if you are using the premium version. Many of the cycling apps disable many of the most wanted features so that if you want them you have to pay. As mentioned, pretty much all the cycling apps have an accompanying website that will allow you to create routes. Some require the premium subscription which gives you full access to both the website and the cycling app. My suggestion would be to try some of the web sites and see which one is the easiest to use and that you like the best. They will all export to .GPX, .TCX format so they are compatible with most of the popular cycling apps. Some may even export to .KLM or .FIT file formats. Also, most of the popular cycling apps share with Strava so that when you end your ride, it is automatically uploaded to your Strava account and displayed on your Activity Feed page. It will be no different than using the Strava app on your phone. I do this with all my rides using Cyclemeter on my iPhone.

Here are a few of the popular cycling web sites that you can use to start with.

Endomondo
Garmin Connect
Map My Ride
Ride With GPS

As for the Garmin, you can create a route using any of the websites but for the newer Garmin units, you still need the Garmin Connect app on your phone to push them on to the computer.