I'm upgrading rims but want to keep the original set. I want to add a new cassette to the new rims. I have SRAM Rival compact. What advantages do the following choices give me on a new SRAM Red cassette?
The 11/23 includes: 11,12,13,14,15,16,17,19,21,23
The 11/26 includes: 11,12,13,14,15,17,19,21,23,26
The 11/28 includes: 11,12,13,14,15,17,19,22,25,28

Most of my rides are hilly with a few triathlons

The ones with lower gears give you lower gearing. If you need lower gearing you might want one of them.

Describing your rides as "hilly" is useless. Some people consider a freeway overpass a serious climb, others think that anything less than 1500' elevation gain is a roller. Even if your hilly is the same as everyone else's hilly, there's big differences in fitness and in riding style.

Use your current cassette as a comparison. If you have been wishing for a lower gear, get a cassette with a lower gear. If you never use your largest cog, get a cassette that doesn't go that low.

Without checking, I thought the SRAM 11-28 had a different 9th cog. I thought is was 24, making a 4 tooth jump. They also go out to 32 with a 12-32. I think the last jump there is 24 to 32. Anyway, check their website. I believe the 32 is available only in Force and Rival grade.

IMHO they all suck because they have at least one and maybe two tiny little cogs you can't make good use of (professionals used to ride a 52x13 big gear) and no 18 cog which feels real good on the big ring and except with the 11-23 you don't even get a 16 cog which is rather nice down-hill/sprinting cog on the big ring or gear for a little head wind or incline on a 39 ring.

As a fit not fat rider I never needed anything bigger than 34x23 to get over anything in the Rocky Mountains (including the Mike Horgan Memorial Hill Climb Magnolia Road route, although I did puke on that one) which is the same as 39x26. I also never wanted anything bigger than a 50x13.

With 10 cogs I'll do 12-13-14-15-16-17-18-19-21-23 (I don't need the 12, but it gives me more top end on my small ring) or 13-14-15-16-17-18-19-21-23-26 (The 26 is only nice with my fat belly on climbs at 6%+, but it gives a better chain line on the 18/19/21 cogs which I spend most of my time on with flatter terrain).

This all depends on what rings you're running, aerodynamics (aerobars vs road drops make a big difference), your mean maximal power output for various durations, how heavy you are, how steep and long your hills are, what your preferred cadence range is, what happens in terms of fatigue when you need to pedal faster or slower, etc.

All of that is personal and makes the question something you need to answer for yourself.

You should know where you bounce between cogs where neither is right and you'd like one in the middle, where you need a lower gear to make it up something without unacceptable fatigue, where you'd like a smaller gear, etc.

For instance I noted that although happy to pedal down to 80 RPM I could ride threshold intervals on back to back days if I kept pedaling over 90 RPM but not at lower speeds and I'm happier doing sustained and repeated efforts at that power level on the low side of 100 RPM. Small cogs running 17-18-19 make that possible with less frenzied shifting. _Training and Racing with a Power Meter_ has an anecdote about a racer who got dropped anytime he spent more than five minutes a power he could otherwise maintain for an hour but cadence under 70. Bigger cogs fixed his problem.

The best answer may also involve learning to spin faster and smaller gears.

Play around with this: https://www.gear-calculator.com/#

Gives you a nice graphic showing how the range, spacing and cross overs compare with the different cassettes.

There's also Sheldon's calculator: https://www.sheldonbrown.com/gears/

But I like the first link better because of the graphic display.

Ericm979 pretty much covered it. Don't over think this. If you need easier gears to go up hills, get a cassette with a larger cog than what you have now. If you are fine with what you have, get the same again. BTW, if most of your build is Rival, you probably will not notice any "improvement" with a Red cassette. Only lighter in the wallet.

I agree with Drew that many people (not all) would be better off with a 12 instead of 11, and add an extra cog in the middle where it will get used more frequently.

This is a pretty nice spreadsheet that you can use to enter two setups to compare some ratios. It obviously won't tell you how fit you are to turn the gears, but it gives a nice overview of cog to cog percentage jumps and speed based on gear and rpm. Just have to edit it with your inner/outer ring sizes (in red), edit the cog teeth column, and then edit the rpm to see how fast you'd be going. It assumes a 700c road tire.
10sp_cassette_jump_chart.xls.zip

Thanks - great education!

i think i'll stay with the 11 - 26 as i like the 23 for the majority of the climb and fall back on the 26 to get me to the top. The 28 sounds nice but maybe too slow, i'll have to tough it out.

Hilly to me is 7 - 9% grades on average, i checked the signs for the trucks this morning.

RED will save me some weight so it seems smart for the extra $$

An 11-25 OG-1090 "Red" cassette weighs 161g with lock ring (weight weenies didn't have a measured weight for an 11-26). The lowest of the 3 shopping results returned by a Google search was $163.

An 11-26 OG-1070 cassette weighs 231g with lock ring. The best of five Google results was $51.

Assuming you weigh 140 pounds (like a 5'10" climber should) and are riding a 6.870 kg bike within striking distance of the UCI minimum the Red cassette will make you a full 0.099% faster up the steepest hills. The results will be proportionally less with a heavier rider and bike.

If you are currently competitive as a climber and plan on racing off the front to an up-hill mountain finish that will give you 3.6 seconds an hour on the chasing peleton and the extra $100+ is well spent every few years.

If you'd just like a shiny bauble (I admit to having carbon fiber cranks, carbon fiber and titanium in my derailleurs, a titanium seat post, etc. They make nice guy jewelry which costs far less than something pretty with an engine. I also get the least expensive Campagnolo cassettes that still come with nickel chrome finish so I don't feel bad dropping them in the trash can) that's fine too but be honest about it.

Otherwise it's not so smart.

Drew, Drew, Drew ...... there you go again. Spouting facts & common sense in an attempt to trick people into saving money.
As Homer Simpson said, you can use facts to prove just about anything !!

Lower gears are not slower unless you're spun out (in which case you should just upshift). The fastest gear will be the best gear. Toughing it out normally works fine if you have several thousand feet of climbing. But if you're one of those types who wants to climb 15K feet or more in one day, you can burn up your spinning muscles and will be in for a lot of hurt. 7-9% isn't that bad, but it's worse than it sounds if you're out all day.

The 11-26 will combine nicely with the compact. 11-28 is actually the same as the 11-26 except at the bottom of the range, so it's also a good option. I don't think 11-23 is a good idea with a compact for the simple reason that there are not enough overlapping gear ratios between the rings.

Just throwing it out there, but Ultegra offers a 12-25, which is my favorite all-around cassette. If 25 is a low enough low gear for you, it's a good bet.

12/25 Tooth (12, 13, 14, 15, 16, 17, 19, 21, 23, 25)

Those one-cog gaps in the middle are just heaven for cruising around.

i'd go 11-26, especially with a compact crank. i've been using shimano 11-25 cassettes but it's basically the same. 50x11 is a bigger gear than 53x12 and will almost always give you enough top-end speed, and 34x25 or 26 should be enough to get you up just about any hill. i've used both of those extreme gears in multiple rides and races but almost never felt like i needed a wider range (with the exception of a certain stage race over the summer when i used an 11-28 and still barely made it up the hill).

Since this won't be your first cassette, what do you have now?

The best thing to do is get whatever you don't have now. If you have a tight cassette, get a wide one and vice versa. If you have an all-rounder, go for something wide. The difference between Rival and Red is not nearly as significant as being able to choose the right cassette for the ride.