300 Burnhamthorpe - sounds like somewhere close to the Mississauga Central Library as MCL is 301 Burnhamthorpe.
My suggestions:
1. GO Bus. From Union Station, take westbound Milton line GO Bus which you can drop off at Square one (Duke of York/Rathburn) and take the eastbound bus (same route) back to Union. Walk south for 5 minutes to get to your location.
2. GO Train. From Union Station, take Lakeshore GO Train westbound to Port Credit station. Transfer to Mississauga Transit bus route #19 Hurontario (you can only go north) and exit at Square One [Burnhamthorpe/Hurontario]. (Note - you only pay $0.50 rather than the regular $2.25 if you take the Mississauga Transit bus from GO station and shown the driver your GO ticket.) Walk west for 5 minutes to get to your location.
3. TTC, then MT as the previous poster suggested.
Option 2 is faster (because the train is faster). Option 1 is more direct (one bus and that's it) but not as fast because it goes through various stops to pick up other passengers. IMO, use Option 3 only if necessary because you'll be in for a long ride.
Check out
http://www.gotransit.com for schedule and route details.