VIP Bus takes about 5 hours from Arcade (Chiang Mai main bus terminal - there are 2 so make sure you use the Arcade) to Mae Sai (for the border) or as said about 4 hours for Chiang Rai town centre (note: CR is a province as well as a town, and includes many towns).
Red SongTaews can also be hired for tours directly around CM (about the same price per day as tours) and will then take you were you want, wait outside, and be private for you all day - usually friendly people too. Look for ones with couples in (driver plus wife and sometimes small kids stuffed in between - less likely to get a drunk then).
At the airport (CM), go to the taxi kiosks, book a taxi to your hotel (150baht fixed fair to town per car/not person) - and while you are there take a free tourist map with all the places of interest in town (and some outside of town).
CM is not wilderness though, it is a town (Thailand's second city after Bangkok), but it is surrounded by mountains and jungle and a short trip (even by bicycle) can put you in either.
Three nights might be pushing it though - the trip to CR would take most of the day (and needs to be pre-booked at the bus station - online too, but have to be a member (free) and is in Thai). CM has a lot to see, and 1 day over the CR trip would make it pretty much pointless excursion. Given 3 days, I would suggest missing out CR altogether and just do CM - plenty to fill 3 days!