Trieste airport (TRS)

20 minutes driving to downtown; busses or taxis easily accessible.

Venice (VCE)

This is the best connected airport, especially for long distance flights. 1.5 hours driving to Trieste; trains easily accessible or rideshare (GoOpti)

Treviso (TSF)

Good option for budget airlines within Europe; rental car required.

Ljubljana - Slovenia (LJU)

Small airport but occasionally has good deals. 1.5 hour drive to Trieste; rideshare available (GoOpti)