To use a gate/pathway you have to go to the Items & Objects in the locale mentioned and use the gate/pathway there.
If you got more information about the paths, that haven't been covered here yet, then please add it.
WARNING: Exploring in Ver 2.0 now takes a toll on health and energy.

(see also quests in alphabetical order)
St.Kobe's Day (Day 40). Can only do one of the following paths each PopYear. If you successfully complete the path the Muse will reward you with extra skill in the genre of your choice. You must be 16 years old to use the statue in the temple to enter these paths.
A tribute to Frank Blomdahl, on Day 27 you can go to the Stockholm Graveyard, Skogskyrkogården, and gently stroke his monolith. In his graciousness he grants a boon of 3 XP.
1) You instantaneously get the diary entry and 3XP . . . and that is all.
2) Everybody can use it (children of all ages, VIPs and non-VIPs)
3) There is no health or mood loss
“New style” quests, which are just there to gather things. (see threadID: 792145.1)
You need a bucket and the agriculture skill to get milk from the cow.