I'd like to be able to climb hills better: is it an option to increase the diameters of the largest gears on rear cassette?
Gears are measured in terms of the number of teeth, not the width.
The mechanical advantage is given by the number of teeth on the front chainring divided by the number of teeth at the back.
Often this is then multiplied by the diameter of the tyre, to give a measurement in 'gear inches'.
Your bike looks to have a 14t-28t freewheel. Your rear derailleur is a 'mid cage' model designed for 14-28t freewheel and a 20t front difference, for example a 48-38-28 or a 44-34-24 triple chainset.
You might be able to change the freewheel for a 14t-34t six-speed freewheel, but if you have a triple chainset fitted, then your mid cage derailleur may be beyond its limits. In this case a similar quality 'Tourney' SGS (long cage) derailleur should be quite cheap to buy if it does not work well.
As you have not indicated whether you have a triple, double chainset or a single chainring at front, it's hard to say the main cause of your trouble - for example if you have just one chainring up front, then the 28t small cog is likely to result in a very hard easiest gear, and in that case the problem is at the front, not the back.
However it is quite possible that you do have for example a 48-38-28 chainset with the 14-28 freewheel, which is not a particularly hard gear compared to a modern racing bicycle sold for non-athletes, which will typically have a 50-34t chainset with an 11-34t cog (28/28 is the same gear ratio as 34.34), but your bike is likely to be heavier, so this still might not be enough.
Anyway, switching from 14-28t to 14-34t or 13t-34t freewheel will give you a considerably easier gear, but you might end up needing a new rear derailleur as well, and it would be easier to answer your question knowing about your front chainring(s).