A private transfer service (like Supershuttle in the U.S.) is also a good alternative. Costs are about €25 per person per way. The main advantage is they pick you up at the time you want and drop you off in the augsburg area where you want. 
If you arrive in Munich at the airport then you will need to take the S-Bahn S1 or S8 to the Hauptbahnhof (main train station), a 45 min ride, before you can catch a train bound for Augsburg. At least once per hour a Regional Bahn train leaves from Munich's Hauptbahnhof (Main Train Station) towards Augsburg. The ride last about 50 min. Plan at least 2 h in total from Munich Airport to Augsburg (as you can see taking a bus or private transfer will be much faster). It is best to purchase a Bayern-Ticket (Bavaria Ticket) from one of the ticket machines located in the train station. A Bayern-Ticket allows one (along with up to four other friends) to travel within Bavaria using all regional trains (no InterCity Express) and city transportations (trams, buses, etc.) from the time of purchase until 3 a.m. the following day.
Augsburg has a small but efficient transportation system  with four main Tram (German: Straßenbahn) lines and several buslines which cover the whole urban and suburban areas. These lines run Monday through Sunday from around 5 a.m. to 11:30 p.m. On Thursday, Friday and Saturday nights, a night bus runs from midnight to 3 a.m.
If you purchase a Bayern-Ticket (Bavaria Ticket), you will be covered during your stay in Augsburg and will not be required to purchase additional tickets for using the Augsburg transportation system. However, if you did not purchase a Bayern-Ticket then you must purchase a ticket and stamp it each time you enter a streetcar or bus. Riding without a ticket (German: schwarzfahren, "black driving") will get you a €40 fee and a visit with the police.
You can also buy a 24 h ticket, a single ticket, or a family ticket.
The Bayern Ticket is only valid after 9AM (Weekdays) to the airport from Augsburg Hbf if you are traveling to the airport. Also, if traveling from Munich Airport to Augsburg, it may be possible to use a Bayern-Ticket Nacht (Bavaria Night Ticket), is valid Monday to Thursday from 6 p.m. to 6 a.m. the next day, Monday and Friday, from 6 p.m. until 7 a.m if you book online for €22, at the ticket machine or at the Reisezentrum (Travel Centre) for €24.
Since Augsburg's topography is not that easy to get, but actually everything is quite near together, it is a great idea to explore this little city by bike. That way you'll have a little more time to get lost and then find out where you are again, instead of waiting for buses or walking around.
Also, a bike can be used for a little trip into Augsburg's Siebentischwald, a big forest and recreational area with many trails going through it, which reaches almost into the inner parts of the city. Augsburg's Zoo is also located on the edge of Siebentischwald, and thus can be visited easily by bike.
Bikes can be rented on several bus stations from the "Stadtwerke", and at some bike stores in Augsburg: Zweirad Bäuml, Jakoberstr. 70, Augsburg; UNI - RAD, Salomon-Idler-Str. 26, 86159 Augsburg; GEPE Sportgeräteverleih UG, Reisingerstr. 27, 86159 Augsburg
Bertolt Brecht Haus The birth place of the writer. Auf dem Rain 7
Fuggerei The Fuggerei is the world's oldest welfare housing and dates back to 1516. 160 people still live in this complex. The rent of an apartment today is only one euro.
Fugger Stadtpalast This 16th century renaissance building is the former residence of Jakob Fugger. Maximilianstraße 36 - 38
Maximilianmuseum Displays verious topics connected with Augsburg. Fuggerplatz 1 http://www.kunstsammlungen-museen.augsburg.de
Rathaus City hall built in the 17th century by Elias Holl.
Schaetzlerpalais Displays art from baroque and rococo eras. Maximilianstraße 46 http://www.kunstsammlungen-museen.augsburg.de
Augsburg Zoo Old zoo of medium size.
Mozarthaus Leopold Mozart's birthplace. Now this house hosts an interesting museum which is dedicated to Leopold Mozart and his son Wolfgang A Mozart.
Augsburg is a lively University town, a city for business and recreation. Enjoy the benefits like traditional pubs, theatre, museums and lots of shopping facilities. You find many shops in the pedestrian zone, but don´t miss to go downhill to the "Altstadt" with its fancy shops! There is also a mall called "City Galerie", common brands, big stores, but in the weekend it is a noisy place. Regarding souvenirs go to the Rathausplatz, find Augsburg-umbrellas and Brecht-T-shirts. The local brewery Thorbräu sells giftboxes with 6 different types of beer!
Augsburg is famous for its plum cake called Zwetschgendatschi.
The Stadtmarkt or city market, enter from Fuggerstrasse or Annastrasse, has fresh fruits or vegetables as well as the Fleischhalle and Viktualienhalle, which have various budget options available, including typical Bavarian, Greek, Thai, sushi and more. A standard for the locals.
A good part of Augsburg's night life takes place in Maximilianstraße and the historic district right beneath the Maximilianstraße. There is great diversity in the type of clubs/bars, ranging from disco to student bars. It is a great idea to go see more than one place and find out which is the most comforting to you.
Beer gardens: In Augsburg there are several beer gardens, and although they are smaller and may lack some facilities like a children's playground, what is written in the Munich travel guide about beer gardens also applies to Augsburg's beer gardens (Link to Beer gardens and beer halls in Munich travelguide)
Gay scene: Augsburg has a fairly small gay scene, but there is one gay bar in particular, the Fegefeuer, which is about 5-10 minutes by foot from the city hall/Maximilianstraße (Ludwigstr. 34, 86152 Augsburg)
Augsburg has some cheap places to stay. One is the Jugendherberge the other is a private hostel called Übernacht Hostel (http://www.uebernacht-hostel.de). The Jugendherberge (http://www.jugendherberge.de/jh/bayern/augsburg/index.shtml.de) is a little Outside of the city center (5 Minutes walk to the town center). The Übernacht Hostel is near the town hall.
Augsburg is a very safe city, maybe nearly as safe as Munich. Virtually all crimes are domestic violence, drug abuse or drunk people getting in a fight at a bar, so no dangers for tourists, except if you get drunk and then get into a fight. There is a drug scene in Augsburg, which has been a little more spread over the city after the police wiped them out of the Königsplatz. However, if you are not behaving very stupid (showing off as a tourist & having an easy pickable wallet or alike), you will not be bothered. Augsburg also tends to be pretty safe at night. Don't be afraid of using the small and poorly lighted streets in the historic district, since this is one of the best areas in the city!