Enjoy fresh sushi or tamago at stalls and small sushi counters in the Outer Market; many stalls open from ~5:00–14:00 so mornings are best. Great way to start with real Tokyo seafood culture.
Take the Tokaido Shinkansen to Odawara (~35–40 minutes by Kodama/Hikari); transfer to local transport for Hakone. Check timetable for every 30–60 min departures.
Take the sightseeing 'pirate ship' on Lake Ashi (service ~9:00–16:00 depending on season) and visit lakeside Hakone Shrine; great views with clear weather.
Ride the ropeway (if open; check wind closures) to see volcanic steam vents — Owakudani hours vary but ropeway often 9:00–16:00; black eggs are a novelty.