As I improve my ship, should I sell off the lesser improvements. For instance, I just moved up from hardened titanium to whipple - is it better to keep both (does it make the ship stronger) or should I just sell off the titanium to free up room for improvements in other areas?

sell them off, install the best modules you can afford but watch the maintenance costs

generally speaking, sell and swap
but I won't do it every tier(or every mark of a tier), especially on the lower tiers. u might wanna plan ahead and do 2-3 tiers in one go and switch after u r done. At early lvl that doesn't really have much difference, but it can save u a lot later.

also keep in mind there are two different types of defense modules (dampeners and thrusters) and two different types of stealth modules (jammers and cloaks)... you can have 4 of each defense category and 2 of each stealth category