The S5/S5X takes you to Leipzig Hbf in 14 minutes.
From there it's either a 10 minute walk, as advised, or one stop to Augustusplatz on any of the trams 4, 7, 10, 11, 12, 15, 16 or 34 (and some others):
![](https://cdn.statically.io/img/i.sstatic.net/2HtpM.jpg)
You can use a single ticket from the airport to anywhere in the inner city, you do not need to get a new ticket for the tram.
You can use Google Maps, Apple Maps or bahn.de to plan your journey.
LEJ airport has free Wifi, as does Leipzig Hbf, and your hotel will probably too. As for mobile internet, it probably makes the most sense to get a prepaid SIM card for your phone; see German SIM card for a two-week visit? There isn't a lot of retail at LEJ, and it doesn't look like any of the shops that exist there are phone shops. However, there's at least a Telekom store and freenet/mobilcom store at Leipzig Hbf, so you can get a card there. If you want more options you can shop around further in the city centre near your hotel, but if you just want to get set up quickly, this is probably it.