The city centre is small and walkable — you can cross the planned core on foot in under fifteen minutes. The town's flatness and broad avenues make cycling especially pleasant, and the bike-share system Gonkolo has stations across both Nova Gorica and Gorizia, so you can cross the border without paperwork.
For the wider area — Solkan Bridge, Kromberk, Brda — you really want a car. Local bus lines reach the main suburbs but are infrequent on weekends. Taxis are available but not cheap. The border is invisible: just walk, drive, or ride across at any of the marked crossings.