I don’t have a ton of experience with museums, but from the ones I’ve been to (Louvre, Hermitage, various American museums) it seems like the displays are skewed toward later medieval and early Renaissance arms and armor. It makes sense, as most people are probably there to see the shiny suits of full plate and the complex-hilted Renaissance swords. Also, we probably just have more surviving samples from the later period.

I like the earlier stuff though, full mail, partial plate, great helms, bascinets and coats of plates. Surcoats, heavy gambesons and other fabric armor. Type X - XVI swords with simpler cross hilts. Any recommendations on the best museums for earlier period medieval stuff, say from 900 – 1350 AD?

these invaluable sets might help with some overview: https://www.flickr.com/photos/98015679@N04/sets/

For weapons, I would strongly recommend the British museum in London. It a very large collection of swords and other weapons spanning from the bronze age to the high middle ages, including a very good collection of viking-age swords and other weapons (and some wonderful slightly earlier stuff like the Sutton Hoo burial artifacts). There is also a dragon-shaped figurehead from a longship, and many other amazing artifacts from that period. The arms and armour of the high middle ages that were on display when I visited were more modest in number, but excellent examples nonetheless: a very large (two-handed) sword of state (I think from Edward III, but can't quite remember), a beautiful and beautifully preserved earlier double-fullered type-X sword with wheel-pommel and mysterious symbols inlayed in yellow metal; some well-preserved pieces of mail armour, and a beautifully painted shield showing a knight and his lady or something similar. There were a lot of other medieval civilian artifacts worth a look, and I haven't even mentioned the bronze age and iron age (Celtic, roman and germanic) artifacts (which are definitely worth looking at but are perhaps outside your own period of interest).

There are also a few older pieces of weaponry and armour in the Kelvingrove in Scotland (especially some older swords), but again that collection is more weighted towards the renaissance and later.

The Metropolitan Museum of Art in New York City.

http://www.metmuseum.org

Thanks everyone for the recommendations. Those photo sets are great and I will be sure to get to London someday. Any other recommendations are appreciated.

You could also try the Wallace Collection in London. Great arms and armoury, some really knockout armour. Any visit to London should include the Wallace if you are interested in medieval arms, but it is a good museum generally.