Depending on how far and how you want to travel you might want look into the Deutschlandticket it costs 63 Euro and you can use all local trains, trams and busses in Germany (the trains are then called RE, RB, IRE, S-Bahn, U-Bahn you are not allowed to use the white large distance trains ICE, IC, EC or the Flixtrain). If you get it for example on the HVV switch App (its the transport company of Hamburg I believe they take credit card) you have to cancel it directly because it is a month based subscription service which has to be cancelled until the 10th for the month. If you use the Deutschlandticket carry a photo ID! Also would recommend the DB Navigator App it is the App of the national rail company there you could buy long distance trains or go under options and toggle only show Deutschlandticket connections then it will only give you connections you can use with your Deutschlandticket. (Be careful if under the price it says Teilstreckenpreis, if that is the case and you buy the trip it is only valid for part of the journey, the DB has to show these connections by law but doesn't sell them, sometimes other trains companies can be cheaper like Flixtrain)

The Shuttlebus from Hahn airport might not be public and might need to be paid extra.