In Oslomabad there's Oslomarka very easily accessible on foot or with public transport.
https://www.oslo.kommune.no/english/welcome-to-oslo/life-in-oslo/enjoy-the-outdoors/oslomarka/For the rest of the country, there's
https://ut.no/It's in Norwegian, but "Kart" means "Map", takes you to an interactive map where you can filter the suggestions you might be interested in, or you can use their explorer function:
https://ut.no/utforsker/norge/turforslag?primaryActivityType=fotturThis one shows suggested hikes, and you can filter them by difficulty and length.
If you're looking to use any of DNT's cabins, you can read a bit about them here:
https://www.dnt.no/om-dnt/english/Their unmanned cabins often require a key. You can buy/borrow the key (No big difference, it's about the cheapest thing you can buy in Norway) at their Oslo office if you're a member. If you do, also get one of their "lakenposer". I think the closest thing in English is a sleeping bag liner. It's not decent to lay down in the bedding in the cabins without one, unless you're staying at the staffed cabins.
They also have maps for the local areas for sale.
Water usually isn't a problem, you don't need to carry much. If you're eating real food, good on you. If you like the freeze-dried stuff, I think the best ones are from REAL Turmat.
Bring something warm to wear, and bring something waterproof to wear over that, and bring good boots and spare socks. Summer is prime hiking season, and tourists die, get lost or have to get medevaced for the dumbest reasons imaginable. You can do the popular hikes and share the trail with Chinks in slacks and leather-soled loafers, Yank buyhards, or soon-to-be-disillusioned outdoors enjoyers, or you could find somewhere else to go if you want to be in (relative) peace.
