Hello !
Normally you can find flats that are furnished or unfurnished but I never have seen any that was inclusive of all the bills. So you will have to add the costs of : council tax (around £100/month depending of area/band, electricity & gas, tv license, insurance, cable/internet/phone)
I know only Didsbury and Salford Quays both have good public transports (bus, tram) and are quite close to Manchester city centre. Didsbury has a large student community and young families. Salford Quays is more for young professionals working in the city (single or young couples). Salford Quays is not that big area and is directly surrounded by areas that are not too safe. Disbury is bigger in terms of area and has nice shops/coffee places/bars. If it was me, I would prefer to live in Didsbury (compared to Salford Quays). I do not know the other areas.
I might be difficult for you to rent if you do not have any credit rating in the UK. Normally they would credit check you and if they can't they might event ask you for 6 month payments in advance and really good references (from your employer,etc..)
It's not an obligation to be with an agent but most flat are rented through estate agents so you will have to contact them if you are interested in a specific flat. You can find flats to rent on here : UK's number one property website for properties for sale and to rent
You will only be able to start your search once you are in the UK, you will find that the estate agents will probably be interested until you are physically here. However you can start looking at specific areas where you would like to live so it doesn't take too much time finding a place once your are here.
hope this helps
Joanne