I've recently got into road cycling and I'm not particularly strong yet but I absolutely love climbing.
However, I'm very short and thin (5'7" and 130 pounds) and on climbs that average 9%, my cadence is about 60 (at threshold) whilst my usual cadence is about 100-105. After 3km, my legs are completely cooked.
I'm running a 54-39 (or 53?) and 11-28 at the back.

I'm tossing up if I should be getting a compact chainring or semi compact, or if I just need to train harder and stop being a sook.

I'm aiming to do some climbs that have 1km+ of 20% but at least 50% of my training is on flat.

Any suggestions on what I should change to?

Get a 50-34 crankset and a 11-34 cassette.

Get 50/34 crankset first an see how that works out. If needed swap the cassette out as 10wheels said.

IMO when climbing, it is your lungs that should be the limiting factor, not your legs. For someone who spins as fast as you do on the flats, a lower gear ratio for climbing would make sense. It has nothing to do with toughness. You aren't a masher, and you need a setup that accounts for that.

Compacts are great for everyday riding and training... and racing. I have one on my everyday bike, but I do have a standard on my race bike so I can seem more badass at a mass start.

A 34-28 lowest gear would allow you to go as fast as you are at 60rpms on the 39-28 with a cadence of ~68rpms. Up to you to decide whether that is worth it or not.

Play around with this to figure out what combination would work best for you:

HTML5 Gear Calculator

Unless you can max out a 52x11, you might as well get a compact with a 11-28 and have more gear options.

My son is looking to swap out is standard crank and 11-25 for a compact and 11-28 since it's not actively racing.

While you are suffering and trying to turn the cranks over 39x28 do you ever wish you had an extra gear?

That's what the compact enables.

Nothing saps energy like trying to grind up a long hill at 60.

Only time I can max out my gears is on a descent. I don't think I could max them out in a sprint yet, or ever haha.
I think I'll go for the 50-34

Appreciate all of your help! Thank you!

I would also go compact, but keep the 11-28 cassette. His rear derailleur might not accept a 34t cog.

Compact with an 11/32 works well for for me. Given an option I would have spec'd out an 11/28 but 11/32 is what came with the bike. I'd originally figured to change out the cassette after purchase but turns out I'm good with the gear spacing and the 11/32 allows me to spend most all of my time on the big ring without cross chaining. And there's plenty lower gears for when and if I do serious hills.

Short cage is good to a 28. You need a mid cage for 30-36.

Exactly why I mentioned it.

I'll do the obligatory.. 'have you considered a 52/36 with a 11-32 cassette" question..

That is a good combination.

I ride a mid compact 52/36 with 11/28 or an 11-27 (Winter wheelset)and love it. Its the best of both worlds for me.

Go for the compact. There are very few times in my life when I wished for a bigger gear and many many times when I wished for a lower one.

Go straight for the 50/34 and save the 53/39 on a shelf in case in want it again when you're in better shape.

choose gearing based upon the terrain you ride and your strength.
This can even mean custom gearing...I ride 50-38 on one bike...and even custom cassette where you combine two cassettes to choose the cogs you like.

As a general rule you want to keep gear spacing as close as possible and yet cover the range in gearing for both climbing and highest descending speed.

I don't think so unless you live in the mountains. Nobody needs 52-11. 50-11 is a tall enough gear for most unless a low CAT racer.

Wut?

I hate to be the contrarian, but I don't see why it's urgent that he should change the 53-39. His 60 cadence isn't all that problematic on a hard climb, but you could gain a little (for a lot less) if you can find a 12-30 cassette. A short cage Shimano RD might (likely could) be pushed that far, if not they're not that much if we're talking 8- or 9- speed.

I just wouldn't want to push him into a compact if it wasn't really needed. With his size and room for improvement ("recently got into" it) he might be spinning up that grade in a few months.

50x11 is good enough for 43mph

Campag is right. 50x11 at 28.5 MPH at 80 RPM. (43 MPH at 120 RPM) To maintain that 28.5 MPH pace, in the drops, on flat land with no wind for a 155 lb rider and 20 lb bike, they'd have to output >400W. That's roughly 6 W/kg. That puts them in the Cat 1-2 - World Class on this chart. So unless you're constantly riding down massive straight grades, anything higher is unnecessary. Even then, your gearing is better spent where you'll be riding the majority of the time, not when you're flying down a hill.

Well, a 9% average grade hill is steep.
At your 39-28 low gear, 60 rpm, that's 6.5 mph.

VAM is vertical meters per hour. (Very steep hills tend to bump up the VAM score, since riders are limited how slow they can go.)
your 9%*5280 feet per mile * 6.5 miles in an hour, converted to meters = 940 VAM

That's a very hard effort, and pretty impressive. Can you dial back the power on each pedal stroke a little, and go longer? Lightweight riders supposedly can climb standing up more efficiently than heavy riders.

On the other hand, a long 20% grade can use all the lowest gears you got. I used to climb a local 120 foot high, 16-18% grade at 3 mph, 30 rpm, with a 34-27. I could do that effort for that distance without blowing up.

VAM Effort (According to me)
400 Easy climbing
550 Pacing up the long climb
750 Working hard, shorter climbs
1000 All out
1200 Race climbers
1500 Pro level