It really depends on a lot of things. Most people stay a few days in Kathmandu upon arriving and then go to Pokhara or Chitwan for trekking/adventure sports and then come back and do some exploration of the Valley like Bhaktapur/Changunarayan. I like to suggest first staying outside of the city in order to get over jetlag. Do some of your souvenir shopping in villages like Changunarayan, where I happen to live, or Bhaktapur, Kirtipur, Pharphing, etc. You can enjoy genuine, Nepali culture and hospitality in such places and often get better prices on thangka paintings, masks, healing bowls and such.
You can plan to spend from $10 per night for a room and $10 for good food prepared properly. You can cut a dollar or two if you just have mo:mo or samosa for lunch and Nepali Dal Bhat for dinner. You can get a good tourist bus to Chitwan or Pokhara for $12-15 for a nice tourist bus; some even have toilets onboard and WIFI. They all stop at a nice place for a breakfast buffet.
Many people like to just stay at one place and get to know the culture better. You can stay with local families in an home-stay from about $10 a day, including food. That's pretty basic, but many people love getting to know a family and supporting the local economy. If a person wanted to stay in a guesthouse like ours they can stay for about $20 per day in a bit more upscale room and restaurant quality food and weekly transportation for sightseeing. That would be a weekly or monthly, well-negotiated, discounted rate. Unless you don't mind a lot of air pollution and dust you won't want to stay in Kathmandu for too long. It's best to stay at a bit higher altitude for fresh air and genuine Nepali life. But everyone should spend at least one day in Kathmandu. It's an amazing place.