door knob/lockset for 2nd floor entrance

We need a doorknob / lockset for the 2nd floor entrance to our townhouse, which will be the main entrance for our tenants (who will be on the 3rd floor) and a secondary entrance/exit for us. So basically, it’s a shared door. Obviously I want something secure, but I don’t feel the need to spend a ton of money on it – especially since I don’t really care about looks, and there will be another lockset on each of the interior doors (entrance to the tenant’s apartment and entrance to our apartment). The big thing I do want is to the door to lock immediately when it’s closed – in addition to the deadbolt. (I can trust that we’ll remember to lock the deadbolt everytime, but cannot trust our tenants to do that…Hence the reason for wanting this feature.) I know this immediate-lock isn’t as good as a deadbolt, but at least it’s something, in case someone forgets to lock the deadbolt. I’m really confused about the best thing to buy – so many options. Any thoughts?