"Å gå (på) tur" works both with and without the preposition, and the two options see about equal use.
This sentence totally confuses me grammatically. 'Pleier' is the verb, which means 'usually'? That's an adverb isn't it, like 'often'? I don't understand 'usually' being a verb.
Some languages, e.g. Norwegian or German, have a special verb with the meaning "doing something usually". In Norwegian this is "pleie", in German "pflegen". The sentence then is constructed in such a way that this verb is followed by the infinitive form of the verb denoting the specific activity in question (here: "to go"). Since English does not have such a verb, you can only circumscribe the meaning by adding "usually", which is indeed an adverb.
You can also say "I tend to go for (on) walks on Sundays", which has the same meaning but makes way more sense for me personally because it actually includes a verb.
Is "å pleie" the same as "att bruka" in Swedish?
Jag brukar studera varje dag - Jeg pleier å studere hver dag?
In some dialects 'å bruke' is used instead of 'å pleie', and it's even acceptable in Bokmål:
"Jeg bruker å studere hver dag" = "Jeg pleier å studere hver dag"
However, I think 'pleier' is more common, so I'd recommend using that.
the "å " is used to form the infinitive, like "to" in English. The construction of the Norwegian sentence is more like "I use to go for a walk", that's why you need the "å ".