Keeping in mind that I am not Buddhist and therefore cannot comment on the religious importance of this area, i will focus only on architecture and natural beauty.
Although commonly referred to as simply Doi Suthep (as I just did in the title) the temple itself is actually "Wat Phra That Doi Suthep." Doi Suthep is actually the name of the mountain that the temple itself is located on. However, if you simply refer to it as "Doi Suthep," everyone will know you are referring to the temple and not the mountain in a general sense.
This temple complex (like most temples) is just a wonderful experience. It is also located in an area with an unobstructed view of the valley below, which just happens to be the entire Chiang Mai metro area.
While the temple itself is quite elevated, you are able to park at the bottom of some stairs which may still be daunting to some visitors as there are a lot of them. I love the way that this primary entrance has banisters adorned with giant dragons from the bottom all the way to the main temple complex at the top (again, I apologize if this is the incorrect terminology.)
Once you arrive at the top, provided people are behaving themselves, it is an extremely relaxing environment. It smells wonderful because of the incense and candles, and the breeze and silence is extremely refreshing. There are signs everywhere reminding people to keep quiet (as is the custom in temples, i know that much) and they actually have staff that will reprimand tour groups for stepping out of line. I normally find this sort of behavior despicable because even if you are unfamiliar with the religious traditions, you are probably aware that shouting or allowing your kids to run around going mental, kind of goes against the "rules" of any religious facility. Thankfully, i didn't see much of this while i was there.
All in all, i would say that visiting Doi Suthep is something you absolutely should do while staying in Chiang Mai especially because the price of entry is so low (it's free.) Do the right thing though, while you are there donate to the temple a bit because the maintenance cost of such a massive and beautiful structure likely costs quite a lot.