The two main must-see tourist sites are within walking distance from your hotel.
Have milk tea in the street, and order chicken momo for breakfast (or you can wait to eat until 11am like the locals and have dahlbat. Be sure to eat with your hands!!)
Walk the 1.5 km to Durbar Square - go early in the morning to avoid crowds.
Also, the earlier you go, the cleaner to air will be (and you'll enjoy a moment of zen - Kathmandu is very quiet in the early mornings)
Durbar Square is probably the most iconic complex in Kathmandu - it's filled with temples, pagodas, shrines, a museum, etc etc. Be sure to bring your camera and don't miss the sort of creepy statue in the south corner of the complex - it's a statue of this girl who was sacrificed to the gods to become a Hindu deity (not sure if that really happened but it's definitely possible!)
I recommend lunch on the street. If you end up in a Western restaurant you'll pay a lot more but depending on how adventurous you are, you might want some familar food. Yak cheese balls are DELICICIOUS, you can find them most anywhere. Dried buffalo is also not to be missed.
Also, better to eat off site (not at the complex) unless you want to pay 2x as much.
Swayambhunath: The Monkey Temple
In the afternoon, head to The Monkey Temple (you can also save this for another day if it's too much, but I want you to have time for the day trip I have planned, which is technically an overnight trip).
You can definitely take a taxi as its 3.4 km from Thamel, but I walked and it was a very interesting walk so I recommend you do the same.
It's also pretty much straight West of Thamel so easy to find, and the walk takes you straight up this amazing mountain that gets steeper and steeper.
First you'll cross the Bismati River -
Keep heading West and you'll reach a temple that overlooks the city - this is NOT the monkey temple but is still really cool!
Once you start to see restaurants with English signs and a huge set of hundreds of stairs, you know you've arrived at "Swayumbanath" (aka 'the monkey temple.).
There are prayer wheels lining the bottom steps of east side of the temple - you can spin each one and make a wish.
As you ascend the steps you'll encounter begging children, women selling crafts and fruit, and all sorts of interesting statues and mini-temples.
Take your time on the walk up (again, the altitude) but also b/c there is a lot to see.
Oh, did I mention the monkeys??
They are EVERYWHERE and really tame, so don't be surprised if they come right up to you! (but I wouldn't recommend petting them b/c they're still definitely wild!)
You have to buy a ticket on the way up - look for a little ticket taker on your left as you ascend the stairs (tickets are around $2, and are more expensive for foreigners than locals.)
The top of the temple will offer stunning views of the entire city of Kathmandu - I came here in the morning and was dissapointed to see a smoggy/hazy view. I was told the view is much better in the afternoon, towards sunset (not sure why, but try going around 3 or 4pm and you should have better luck).
There's lots to explore at the top besides the view, more pagods, temples, statues, and monuments, plus when I was there I saw a crew shooting a Nepali music video and a medicine man performing blessings on newlywed couples.
In the evening, go see some live music back in Thamel and enjoy some Gorkha at happy hour (watch out though - if the altitude is effecting you, the beer will go straight to your head!!)