The main factors that will influence your bike choice are:
your size, how tall you are, what size bike you need.
Your riding style, how hilly your area is, how far you need to ride, how hard you want to pedal.
Someone that is riding 2 miles, on flat terrain, mostly in the dry does not need as fancy a bike as someone that rides 10 miles, with hills, to get to work or school, raining or not.
Some road bikes are race bikes. That only take very skinny tires and have hard, fast gearing.
Some road bikes are touring bikes or cyclocross bikes, that take fatter tires and fenders, have wider range gearing, and fit racks to carry stuff.
Under a $100, your options are limited. Your best bet is to find a 1980's japanese touring bike or road bike. Univega, Bridgestone, Nishiki, Miyata, Centurion, Shogun, Fuji are good brands to look for. These are higher end 10 speed brands, and realistically go for $200 most of the time, but sometimes you can find deals on CL. Trek made good american bikes, and Raleigh good british bikes. I hesitate to recommend french road bikes, like Motobecane or Peugeot, or most Schwinn 10 speeds, but that is another topic.
Here's a good example of a $100 japanese 10 speed.
It has 27" wheels, which take up to 1 1/4" tires. Current road bikes have 700c wheels. :/ The bigger, 27" wheels are a bit heavier, than smaller 700c, but roll over bumps easier, and fit fatter tires and fenders more easily than a lot of 700c bikes.
It has a 10 speed drive train, 2 gears in front, 5 in the back. The gearing is pretty hard and fast. Not really any super low, easy hill climbing gears.
The schwalbe HS129, $16, is the best affordable choice.
The gears on the bike are 2X5, for 10 speeds. 52X40 in the front, 14-28 in the back. This bike does have suntour bar end shifters, which are pretty cool. You don't have to take your hands off the bars to shift. You do have to be "in the drops" to shift though.
Most ten speeds from this era are going to have downtube shifters or stem shifters.
It should be noted, that stem shifters are very inexpensive, and is pretty much the cheapest way to convert an old ten speed with downtube shifters, to get the shifters up to your bars.
The 2X5 gearing on bikes this old, is not very wide. 52X14 is your top gear. 40X28 is low gear. Modern bikes with triple rings or compact cranks will have 34X32 low gear, maybe 4 or 5 gears lower than old 10-speed.
Also, old bikes like this, have old fashioned chains and cog tooth profiles. Shimano came out with HG technology chains and cogs with cutaways, that make shifting way smoother, and chains quieter and smoother, in the late 80's.
One of the best things you can do for an old 10-speed is replace the chain and freewheel. $8 for KMC chain, $15 for shimano freewheel. $25 to make your bike shift way smoother, and better under load, and increase your gear range is pretty good.
I am going to show you a crappier 10 speed, for comparison to the nicer quality bike, so you know what to avoid, and what to look for.
Poor quality steel cottered cranks:
Better quality aluminum cranks on the Nishiki. These are not super high end, decent quality bike:
You can also see higher quality fork lugs, frame lugs, and paint of the midrange Nishiki:
Another give away of a lower quality bike, is stem shifters and suicide levers:
Getting back to the Nishiki.
So you've got your $100 japanese 10-speed, and have put $30-60 worth of good tires into it. It's pretty fast and smooth riding, but the brakes don't work that great, and the shifting is kinda clunky and loud, and doesn't have very low hill climbing gears.
The brake levers on this bike, don't have rubber hoods, have exposed cables, and not that good leverage. Modern brake levers would be a big upgrade, both in braking power, and ride comfort.
$23 Tektros work. Shimanos are $30, or you could look around for used nicer ones.
The brake calipers on the $100 Nishiki, are aluminum, single pivot, Diacompe 500's, A decent quality brake in 1981.
However, modern, shimano dual pivot brakes are much stronger, and easier to adjust, and cost $15 per side, hardly more than the $8-10 new brake pads cost. The main thing you need to make sure you get right, when ordering new brakes, is the caliper reach, that is, the distance between center of bolt the mounts the brake, and center of the brake pad where it hits the rim. There are 3 differnet sizes, short, medium, long.
Pretty sure this nishiki needs short, 39-49mm reach brakes, like this $15 Shimano Soras.
To me, having brakes you can depend on for ~$50 is worth it. Plus more comfy hoods, where you spend most of your time riding.
The best old 10 speed brakes are the type like this mafac racers. Weinman and Diacompe also made similar brakes. These are quite strong, and new brake pads will have them working about the same as new dual pivots.
Lastly, is the gears. The old fashioned teeth on freewheel and old style chain, make a racket, and shift poorly. Just putting a $8 KMC Z50 chain on will make a big difference.
Putting a $13 shimano freewheel on, with HG shift cut outs, makes shifting WAAAAY smoother.
This what the gear ratio chart for a current road bike, with 50/34 compact crank, and 11 speed 11-32 gearing looks like. There are 3 easier gears, going to down to 2.1% low gear. And 3 harder fast gears, going up to 9.2% And the jumps between gears is smaller.