The proper name for your cluster is "freewheel." They screw on. Cassettes slide on. New tech. IMO the largest advance in bike technology has been in tires, same as with other vehicles. So a new bike not only gets you more and lower gears, it also gets you access to modern tires.
Modern bikes usually come with a "compact" crankset which is 50/34 instead of the older 52/39. So that helps right there. If you look on Craigslist and other places used bike are advertised, you can probably find a 9 or 10-speed triple. That'll have 9 or 10 cogs on the cassette and three chainrings, so in your current bike's nomenclature 27 or 30 speeds. On a triple, the smallest chainring, called the granny, will have 30 teeth.
Bike gearing is measured in gear-inches. Gear-inches X pi =,the amount of advance of the bicycle in one rotation of the cranks. Gear-inches = wheel diameter X chainring teeth/cassette or freewheel teeth. A full explanation of gear-inches is here:
https://en.wikipedia.org/wiki/Gear_inches
You current lowest gear is therefore 27 X 39/34 = 31 gear-inches. If you had a triple with a 34T cog on the cassette, your gear-inches would be 27 X 30/34 = 24 g.i. A compact-equipped bike with that cassette would have 27 X 34/34 = 27 g.i.
Mostly what makes climbing easier is fitness, which is most easily obtained by regular hill climbing. Yes, it's hard and yes it hurts. It takes a few years of consistent riding to get fit. "Consistent" is key, riding several times a week with one ride that's really exhausting, year-round. I started riding again on a bike about like yours at 50. It took me 3 years to get fit enough to ride a century, 100 statute miles.