Mubarakiya, a market stretching back 200 years,  is certainly the best known and most often visited historical/heritage site in Kuwait. Bazaar has a helpful map here. The artist Zahra Marwan made a beautiful pictorial map of Mubarakiya a few years ago, as part of a project with Huda Abdulmughni, which you can see above. Here are some of the sites within Mubarakiya:

1. Safat Square
2. The popular restaurant Shater Abbas and the oldest restaurant in Kuwait
3. The Kiskh or Kiosk of Sheikh Mubarak
4. Historic Water Market
5. Souq Al Manakh
6. Gulf Bank Headquarters
7. Historic Post Office
8. Mubarakiya School
9. The National Library Bookstore
10. Souk Al-Kabir Mosque

Here is a wonderful “then and now” of New Street from the instagram of Ali al Rais. He has posted more old photos of Mubarakiya here and here. The drawing comes from The Kuwait Urbanization by Saba George Shiber.