Here's the thing, in stock, you'll be up against MINI Coopers and Honda Civic Si's and such like.
Doing some simple upgrades, will put you in a street prepared class such as adding, or making your sway bar thicker.
Any increase in your wheel width past 20% of stock will do so also.
Most of the time, most other drivers won't mind if you stay in stock class with mods, until you beat them.
You can simply add tires as David mentioned, and do nothing else will get you best the chance. Put somewhere around 38-42 psi in yer tires and have fun.
Chalk them to see if you get any roll.
Top up your shocks and check your tie-rods and wheel bearings for wear.
Sway bar, poly bushings and end-links will get you most instant gratification, along with tires, BUT, things will wear out very quickly if your Sprite isn't in good condition.
Tech inspections will look for battery hold-downs, throttle return springs, brakes (NO LEAKS, good pressure) and suspension and steering.
Any other leaks from the car will usually fail it as well, because other drivers don't want to drive through our much loved ooze, no matter where it comes from, engine, tranny, diff or radiator.
HAVE FUN!! It's addictive, and it leads to other things.