26" is a tricky size, there's something like 5 different 26" that aren't interchangeable.
And being a department store bike, there isn't much in the way of specification posted.
Before you go tire shopping, try finding a tire size in millimeters.
Most common 26" is 559mm Bead Seat Diameter( where rim and tire overlap).
If you can find that, you're good to go. Plenty of nice tires in that size.
I like the Continental Sport Contact in either 1.1 or 1.3 width.
Narrower tires won't "give" the bike more speed. Speed still comes from YOU, the motor.
Narrower, lighter tires pumped firm will likely redure the rolling resistance somewhat.
Which is what allows YOUR input to create more speed.
Mostly, lighter wheels will spin up faster.
Doesn't matter much when moving, but has a big impact on feel when you bring the bike up to speed.
And "adjusting the gears"...
All Schwinn Huntington Women's bike 26" that come up at a quick search are singe-speeds.
You can adjust the GEARING, but NOT the "gears".
Whether there's any benefit to be gained from that is anybody's guess, since the stock configuration isn't known.
Or your issues with it.
And "go as fast as possible"...
All engines - incl humans - work best in a (sometimes tiny) range of resistance and rate-of-turn.
If you feel that the pedals are flying away from you , and your feet can't keep up - then you've "spun out" and MIGHT benefit from a taller gearing - bigger chainring(up front) or/and smaller sprocket(rear).
If you feel that you're pushing too hard(stalled out), then a lower gearing might be good for you. Bigger sprocket/smaller chainring.
You want to find the point where while turning the pedals as fast as possible, there's still a bit of resistance to turning them.
Keep in mind that knees generally don't mind bending as such, but pushing HARD while bent will often upset them.
While it may feel unnatural to inexperienced riders, pedalling FASTER is generally a better recipe for more speed than pedalling harder.
It is well agreed that people generally do best at a cadence(rate-of-turn) in the 80-100 rpm range.
If you're not there, work on pedalling technique, not bicycle technology.
And, since yours is a single-speed, you have to pick a compromise.
A tall gear that lets you go real fast will be hopeless for acceleration. Will make the bike very user-unfriendly on anything but open road. You'll grumble, grunt and strain at every intersection. And every climb.
A lower gear will make it easier to get moving, but you will spin out at lower speeds.
Last but not least - don't polish a turd.
This is a very basic bike you're talking about. I see prices of $110-$130.
And it has a very upright riding position.
Even if you get the gearing right for spirited riding, air drag from that upright position and the difficulty in bracing yourself against the bars will limit your "go-fast-ability"considerably.