The riverside is where you are going to get the best bang for your buck in terms of uninterrupted bike riding without having to worry about scooters and cars.
I’ve explored most of the riverside paths and I still find the best portion of the bike path is the north side of the path from Rainbow bridge (close to Rao He Night Market and Song Shan station), heading west towards Da Zhi Bridge. You got 101 in the background and Yang Ming Shan in front of you. Not to mention the path is very quiet along Neihu Software park area as there’s not much residential buildings in that area, which means less people.
We have this thread for Riverside bike path news
For walks in open spaces and not having to head all the way over to the riverside. I would try Yuan Shan MRT station area. Plenty of open space (actually not too far from the riverside too). The place does get busy on a nice warm sunny day and summer evenings, so keep that in mind.