Mixte has a 14-28, should I go to 13-32. 14-34, or 14-38 to make hills easier?

After my little adventure to the store pulling the kids in the trailer on my Mixte, I'm thinking I might want to swap out the freewheel so I can get a lower hill-climbing gear (I didn't have any trouble getting to the store and back, but anywhere else I'd want to ride with the kids there are much bigger/longer hills). Will the 32 give me enough of a difference from 28, or should I look more seriously at the ones with 34 or 38 teeth? The only reason I ask is the 13-32 may be available cheap, where the others would be significantly more (like $40ish after shipping).

I should get a new chain too if I swap for a NOS freewheel, right? Would I have to worry about anything else? Everything I am looking at is 5 speed except for one ultra 6, which should fit in the same space.

I think my chainrings are 52t and 42t, however I'm not sure on the small one, I lost my place trying to count it twice and it might be 40t? Would it make more sense to replace one or both of the chainrings?

Help, I feel stupid about all of this, I just know I want a lower gear than I have!

This is a very complicated question to answer without knowing anything about your components or your budget.

First - you need to research the chain tooth capacity of your rear derailleur. Since you are talking about 5-6 speed freewheel, I'd guess you have friction shifting but this also means researching your derailleur maybe a little tricky.

Second - you need to determine if your front derailleur and shifters will handle a triple crankset. With friction shifting, you might have the option of a triple crank which would give you better options for dealing with a trailer, weight and hills.

Only then will you know what options you can go with the components and cost.