Reddit mentions of SHIMANO HG400 9 Speed Mountain Bike Cassette - CS-HG400-9 (11-36)

The gears on the back wheel are the cassette (the gears in the front are the chain rings).

A cassette with a wide range of gears has a very small gear (fast but harder to turn) and a very big gear (slow but easy to get up hills). In 9 speed cassettes, a wide range would be 11 teeth through 36 teeth. For example, this one.

If you're struggling with hills, a cassette with a larger gear will make it easier to make it up the hills. Road bikes usually have a smaller range of gears vs. hybrids or mountain bikes ... so to answer your question, no, a road bike probably won't help with getting over hills more easily. Also, a new cassette, at about $25, is easier on your budget. It will require some specialty tools to install so it might make sense to have your local bike shop install it for you (probably another $25).

Ya you either need a new 700c rear wheel or to lace a new rim to the current hub. . Not sure if your bike has a cassette or a freewheel. It says 6 speed gearing so my money is on a 6 speed freewheel. If that’s the case you need a wheel set up for a freewheel and also a freewheel remover if you are gonna do it yourself.

I’m gonna throw an upgrade at you to consider.

29er wheelset (same size as 700c but called 29er because it’s a mountain bike rim.)

Mountain bike wheels are a little beefier and will take more punishment. This one is cheap and comes with tires already.

Mavic Rim 29er Mountain Bike Wheels with Disc Brake Shimano Hubs PLUS Free Continental 29x2.2" Race King Tires and Tubes! https://www.amazon.com/dp/B01HDTR27Y/ref=cm_sw_r_cp_api_i_SlF9AbR7A1F1B

9 speed cassette
Shimano HG400 9 Speed Mountain Bike Cassette - CS-HG400-9 (11-36) https://www.amazon.com/dp/B073CFBPF9/ref=cm_sw_r_cp_api_i_boF9Ab48BVPKH

SRAM X.5 Rear Trigger Shifter for 9 Speed Drivetrains, Black, 9 Speed https://www.amazon.com/dp/B0063HG45Y/ref=cm_sw_r_cp_api_i_bpF9AbA9TRZ94

SRAM X5 Medium Cage Silver 9 Speed Rear Derailleur https://www.amazon.com/dp/B00554231U/ref=cm_sw_r_cp_api_i_QqF9AbC18FV41

KMC KMC023 X9.93 Bicycle Chain (9-Speed, 1/2 x 11/128-Inch, 116L, Silver/Black) https://www.amazon.com/dp/B001AYJF2E/ref=cm_sw_r_cp_api_i_lrF9AbDFWEG5T

So basically $280 would get you new wheels that are tougher and come with bigger cushier tires, and allow you to upgrade to disc brakes in the future. Also we’re scraping the terrible 6 speed freewheel and upgrading to a 9 speed cassette. It will shift better, give you better low end torque and high end speed. The wheel set also allows you to upgrade to 10, 11 or even 12 speed but 9 should be sufficient.