Hamburgs public transport system features a well connected network and consists of busses, rapid trains (S-Bahn), and subways (U-Bahn). The airport as well as the main train stations are all connected to the public transport system. You can check connections, download the public transport app and buy tickets here: HVV. Tickets can also be purchased manually at machines at all S-Bahn or U-Bahn stations.
Don't get confused: Most of the Hamburg U-Bahn run above ground, while the S-Bahn goes below the city and only runs above ground in the suburbs. We recommend you to take HADAG ferries running through the Elbe river once if you have some spare time, which you simply use with your public transport ticket.
If you want to go around more environment-friendly, you could use the Stadt Rad system of rental bikes. Around 120 stations are distributed around the city, clearly visible by red colour. There is even one station on the DESY campus. The first 30 minutes are always for free, which means you could go around for free all day, as long as you check in to any station within 30 minutes after departure. To activate the bikes you need to book one via android or IOS app.
Further alternatives are the IOKO shuttle service (on demand) and the door-to-door car-sharing service MOIA. The first one is integrated well in the public transport system and you may use it with your HVV ticket plus small extra charge, the prices for the second one are between public transport and taxis. For both services, you need to install an app on your mobile phone to book a ride.
Lastly, an overview of the Hamburg taxi services and prices is provided here, bookable by web or phone. Typically the DESY guards are very friendly and will assist you booking a taxi.