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Chiang Mai is a very good base for exploring the North of Thailand.
One-day-treks can be booked all over the town for around 1000 Baht. They will do the standard tour, a little trek, an elephant ride, hill tribe visit and rafting. If that's what you're looking for you'll be fine just booking some tour. I would personally ask for a private tour and pay a little more to get a better experience.
Walking around the old town can be done without a guide. Ask the hotel staff to give you a map and let them show you points of interest. Make sure to visit some of the (night) markets around town. It's all about food.
Chiang Rai is 3-4 hours bus ride. There are buses leaving every 30min from Arcade Bus Station. It is definitely worth visiting.
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I was in that area over twenty years ago, and we were travelling by mini bus in a group of six people. Chiang Mai is bigger than Chang Rai, which however is not a small town. The distance between the two towns is approximately 200 kilometres via the shortest route, and the distance could be covered in three hours. There is another route which is longer (about 270 kilometres), which would take four hours to cover. The second route runs closer to the Myanmar border. Chang Rai is the most famous Thai town in the Golden Triangle. When I visited, there were lots of people dressed in traditional costumes which were there for tourists' sake. You were allowed to take photos of the gateway which led to Myanmar but could not cross over. The Laotian border is not far away, either.
I don't know of public transport in the area, and so far have not found useful links. However, you may find information about both towns on Wikipedia. This is the link for information about Chang Mai whereas here you can find information about Chang Rai.
Upvote:2
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!