You need a better mouthpiece. You don't need to spend a fortune. This topic has been frequently discussed on here producing a range of preferences. My personal recommendation would be for the Bari Esprit II.
While the Selmer S80 is a good mouthpiece which I currently use on Sop and Baritone and have tried on alto and tenor, it is expensive. Add to this the confusion for a beginner of picking the right lay and an expensive mistake is easy to make. The same applies to the yamaha and the rico graftonite which while not being expensive have a bewildering array of sizes that are confusing to a beginner.
The Bari Esprit II has many of the characteristics of the Selmer S80 with little of the cost. It comes in only one lay and so does away with the confusuion of picking the right one. Mouthpiece ligature and cap retails for around £20. Once the embouchure has started to develop and this piece is mastered, the search can begin for a more suitable piece.
As far as reeds go the selmer paris S80 prefers french cut reeds. Vandoren classic, rico royale whereas the rico and yamaha prefer american cut, rico orange box, Vandoren Juno. As far as I'm aware , the Bari has an inbetween lay and will play with both. Another point in its favour imo.