will they increase your sprint or average speed on the flats?

depends on how you climb them

if i can average climbing a hill at 10mph and if i averaged that hill 5mph before, will it improve my mph on the flats?

Getting fitter will make you faster. Obviously.

Heck yeah! It will make you a much stronger rider.

Skinny guys climb well.
Big guys are fast on the flats.

Pick one.

It my case it does appear that a lot of climbing is making me faster. Riding the mountains strengthens the leg muscles and makes it easier to maintain speed through rolling hills and on long low grade inclines. This translates into faster overall times, etc. But I also do sprints, work on increasing cadence, and do one mile intervals in the 24-26 mph range to get faster.

incorrect

This is very hard to dispute at the pro level, but very easy to dispute for typical BF members. Climbing hills makes you a better rider in all aspects. Riding very hard on flat ground helps too.

I am not a typical BF member. I thank the holy god of pcad for that.

I'm big and I climb slowly. I pass climbers on flats all the time. Climb training makes me stronger but not necessarily faster in sprints. Sprint practice makes me faster at sprinting.

I made some assumptions about the fitness & experience level of the OP.

I know some of the 41 actually have skillz. In fact, any moment now I'm expecting someone to post angrily about how they average 24mph uphill yet it doesn't meaningfully improve upon their 40mph sprinting.

Depends on the grade of the hill and your current fitness level, obviously.
If your hill would be 2% and you go only 10mph it won't help a thing.
Training around and above your LT is what makes you stronger.

climbing hills makes you better at climbing hills

The question is the effect of effort level and cadence, not hills vs flats. You can exert high effort at low cadence on flats by gearing up, and low effort at high cadence on hills by gearing down (within the limits conditions and your gearing, of course).

this, ime

I weigh 142lbs, how do I know if I'll be a better climber or sprinter? I ride about 18mph on the flats and I've been riding since the beginning of summer.

Enter a bunch of races, some with climbing, some with pack sprint finishes. See how you do.

At this point you've been riding a couple of months. Just keep riding lots.

If you don't race, it doesn't matter.

If you do race, your results will begin to establish a pattern of strengths and weaknesses,

And if you eventually want to get serious about it, get a power meter, do field tests, and see how your power from 10 seconds, 1 minute, 5 minutes, and an hour compare to others of similiar fitness levels.

So wait ... you've only been riding for a few months and already you ask yourself what type of competing cyclist you want to be? I like your positive thinking
You can't just choose what you are going to be ... it depends on your genetic glass ceiling.
But ... if you've only been riding for a few months you should be still way under that ceiling and so it's going to be hard to tell.
What I advise you to do:
Train on muscular strength through winter on legs, core and upper body ... make sure to sleep and eat enough and take a day of rest or two from time to time ... then come spring start doing interval training.
You can do this outside by doing fast hill repeats or inside on a trainer.
Train well above your current ability but make sure not to hurt yourself either.
Then ... start doing some time trials against yourself: pick a destination some 10 miles from your house and ride that same ride at least once a week as fast as you can ride ... you'll start getting faster soon enough. If you still feel reasonably fit after you arrived ... you simply need to HTFU and ride faster or further.
Do that for a year or two and you might be closer to your potential ... giving you more outlook on what type of rider you are.

your weight on its own is sort of irrelevant. so is your speed on flats...are you riding on the tops or in aero bars? is it windy? too many other factors.

to go fast on hills, you need good power-to-weight ratio. to go fast on flats (at more or less constant speed), you just need lots of power.

at this point, any kind of riding will help. as you get better, do what everyone else has already said - train, race, evaluate, repeat.

Or he could move to Belgium and ride keremisse for a year. <snark>

"keremisse" Hahaha!
It's not written like that but that's indeed what it sounds like in local dialects!

Hill repeats help everything, but not as much as specific workouts for specific skills. Generally, it's good advice for newer riders just to log lots of miles for the first year. Then start mixing in some intervals. If you like where the results are taking you and think you can stick to a plan, build a structured training plan.

Strength training does surprisingly little for bike performance. Planks are good for the core which helps you hold an aero position, but leg exercise won't make you faster and upper body work just adds "non functional" weight. The advantage of strength training in winter is that it adds some weight bearing exercise to your regimen which is good for the bones.

yes