Hi Богданов,
I am glad, that I even may be of use for someone. The next time you visit your daughter, spare some time to make a trip to the Taunus mountains west of Frankfurt. Depending on the area in Frankfurt you start, you may need less than an hour to get to the foot of the Mountain, most likely it is half an hour. There is also a train station with a direct connection (S4) to Frankfurts Main station (about 30 minutes as well). You may exit in the city called Kronberg, where also a professional bike shop is located (5 minutes from the station). It is not their main business, but they do rent bicycles. This is the starting point of one standard tour and a nice partly medieval city. Another recommendable route starts in "Oberursel Hohemark". I can give you some GPX Tracks if you are interested.
There are some locations often integrated into the Taunustrips. Most pass by the "Fuchstanz", which is are two small restaurants in the woods (30-60 Minutes from either Kronberg or the Hohemark). Then the next destination is the "Feldberg Plateau", the highest Landmark, also with some restaurants and also reachable by bus and car. Thus it is full of day trippers, cars and motorcycles. Because the two flow trails start in this area it is still worthwhile. If you are seeking to be in more deserted place, you can head from the "Fuchstanz" to the "Altkönig" (Old-King) which is the third highest landmark and only reachable on smaller paths. No flow trails originate here, but in this part of Germany it is allowed to cycle on minor paths as long as it is no wildlife sanctuary (there are some!) or no sign prohibits cycling. Both destinations (Plateau and Altkönig) are reached within less than an hour from the Fuchstanz (even 20 minutes are manageable).

If you are into cycling quietly, you may find the Feldberg to crowded. In that case, the Odenwald (Approximately 1-2 Hours in the slower trains) or the smaller foothills of the Taunus may be suited. North of Frankfurt road cyclists will enjoy the more rural suburbs.

For a short and relaxing trip, just follow the Rivers "Main" or "Nidda". The paths are in a quite good state and without major slopes. For a longer Trip you can go to the River "Rhein" and follow it downstream. The area is known for wine and castles and not far from Frankfurt.

Богданов, I will send you the gpx tracks taken from the "Mountainbike Magazine" later.
I have some of my favourite tracks, but I need to export and edit them first, so it will take some time.