Hostel Foot Prince is the place to stay. The staff is friendly and very helpful.
There is a few volunteers working on local projects so as a westerner you won't completely feel a out of place.
The hostels mainly take volunteers but anyone can stay. it is affordable and there is a good family atmosphere
The Hostel is owned by local Volunteering Organization call Foot 2 Afrika. Its a place where all people from around the world meet and share their possible way to serve Africans through hands and skills.
The Hostel has private room, double and dom rooms. There is a kitchen and big common living room however you can take a breath with Mt Kilimanjaro view on their garden.
Their cost are $15 per night including dinner and breakfast. If you prefer a single and private room you have to book in advance and the price is $20 per night including dinner and breakfast.
The food was amazing, and I even got some Swahili tips from the staff. There were travellers there to chat with, and plenty of space to also do your own thing if you wanted. Very clean, friendly and a good spot in Moshi.