The house you'd like to buy will have a listing agent (Realtor). To sell the house you have, you will hire an agent. They will work together if necessary, so it's not up to you to hire them both or pay them both.
What is easiest though is to hire one Realtor to find a house for you, while putting your house on the market. If the Realtor finds the perfect house at a price you can afford, you can put in a conditional offer, subject to the sale of your own house. That done, the agent or Realtor will work harder at selling your present house, since the commission will be better. You only pay the commission on your own house, by the way, not on the house you buy. Just thought I'd better add that bit.
As for your current mortgage, the purchaser might be able to assume that, if the mortgage company agrees. If the purchaser prefers to get a new mortgage or pay cash, you must pay off your mortgage and the mortgage holder will usually require a fee to close it out. However, if you're willing to let the same company mortgage your new house, some kind of arrangement might be made regarding lowering any fees.
The Realtor can provide all the information you need for all transactions involved. It may be complicated to us, but they are experienced and know exactly what they are doing.
All the best to you in the big move!