Augsburg is most easily reached via bus or private transfer from Munich Airport (German: München Flughafen).
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. Examples: http://www.bavaria-schoenegger.de/
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), an almost 45 minute 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 forty-five to fifty minutes. Plan at least two hours in total from Munich Airport to Augsburg (as you can see taking a bus or private transfer will be much faster). The best bet is 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 (for example, trams, buses, etc.) from the time of purchase until 3AM the following day. The Bayern-Ticket costs €28 when purchased from the ticket machines and €30 when bought from the counter.
Augsburg has a small but efficient transportation system  with four main streetcar (German: Straßenbahn) lines and several buslines which cover the whole urban and suburban areas. These lines run Monday through Sunday from around 5AM until 11:30 p.m. On Saturday and Sunday evenings a "Night Bus" runs from 12PM-3AM making limited stops along all Augsburg stations.
If you purchase a Bayern-Ticket (Bavaria Ticket) then 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 (1 Euro) 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.
The Bayern Ticket is only valid after 9AM 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 ) which is valid Monday to Thursday from 6 p.m. until 6 a.m. the next day, Monday and Friday from 6 p.m. until 7 a.m if you book online for 20 EUR, at the ticket machine or at the Reisezentrum it is 22 EUR.
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.
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 is famous for its plum cake called Zwetschgendatschi.
If its local flavor and cheap liters you are hunting, Bine's is the best bet. Take the #3 tram toward Inniger stasse and get off at the schertlin strasse stop. take a left and its about a block up on the left. you can also take the #41 bus from konigsplatz to the prinz-karl stop. Bine's is great and the people who own and operate it are friendly to Americans.
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.