When you reach a new level in a tech a list of possible upgrades will pop up.  There is a description for each one and in the description it will list any new weapons or armor.  You can only pick one upgrade on the list.  If you pick an upgrade that has new weapons or armor (or horses or items) it should show up in the store.  You can also upgrade your unit design with the new tech.  You must be in your own territory to go into the store.

The tech description will also tell you which upgrades will give you new places to explore.

I don't remember the name of the upgrades, but generally warfare will give you new weapons and armor and adventure will give you new places to explore.

I hope that helps

There are a couple of bugs where if you do something when the tech chooser pops up, it won't let you pick the tech or the new tech line. If you want new weapons, you can open up the tech window, either from the button on the very bottom or from the button in the kingdom overview browser and select the warfare tech path. This is one way to get new weapons. There is an alternate route which is the quest/goodie hut route. As you level your adventure techs, new quests and goodie huts will emerge, giving the possibility of conferring special weapons and armors like +2 longswords, etc.