Tripsweb planner (Note: this does not always give you the best connections! Sometimes it is best to find how long a single leg will take: enter the precise starting and ending street intersections for the bus/train you are looking at.)
Express buses along LSD to downtown: #6 (7 days a week), X28, 2 (only certain M-F hours). You can connect up with any train line downtown. For example, get off at Jackson to connect to the Red or Blue lines. This can sometimes be great for going to the airport, since you have only one connection (#6 to the blue line).
To Red and Green lines:
Garfield/55th street bus (#55); during M-F rush: X55 (catch at 55th + Lake Park, Woodlawn, or Cottage Grove).
To Red line: #15 bus to 47th street stop.