I bought my CrossCheck about a year ago on CL and this is what it came with:

DA Cranks (older): 52/39
Rear: Shimano Ultegra RD 7700
Front: Shimano DA FC 7701
DA Barcon Shifters
and I think 12-25 cassette.

I am not too bike tech savvy. I can change tires, replace brake pads, and clean and lube the chain. I am slowly trying to learn more, but don't know much.

Anyhow, I need to get lower gearing. I don't do group rides, I am not a racer, I don't even wear the typical outfit. I just like going out and riding. Working my way up to century riding and hopefully a tour down the West coast at some point.

I rode three 40 mile rides this past week and noticed that I spend most of my time in the 52-23 (for me ~15-17 mph). I have been riding mostly flat, just up and down under the overpasses, because I don't have the gears (and legs/lungs) to do the climbs. To do the longer rides that I want I am eventually going to have to do the hills. I also ride a MTB (1x9) that is 32 and 12-34. Short steep climbs I can mash up, longer sustained climbs I do a lot of pushing.

Looking at the Sheldon Brown calculator it says the 52-23 is 61. That is all fine and dandy, but I have no idea what that means. Also, as I have been researching and people talk about 1 step or 1 1/2 step and cross chaining. I don't know what any of that is.

Basically, I need lower gearing. I was looking at either of these:

46/30: https://www.velo-orange.com/grcru50cr.html
or
48/36/26: https://www.velo-orange.com/suoldlotr.html

Then I need a wider cassette, maybe something like on my MTB. From what I have read, that means I need a different RD (long cage?). I don't know about FD?

Gear inches are a way of describing what we call .. an equivalent wheel diameter ..
so 61" gear ratio is as if it were an old penny-farthing bike with a 61 inch diameter drive wheel.
Or a real big Unicycle wheel.. of same diameter .

cross Chain is using the outer large chainring with the innermost large rear cog , and visa versa, the 2 smallest cog chainrings are literally forced into a heavy side flex as it goes around, wears out the chain , and a well thought out gear ratio set has that same ratio in a better chainline.

Ideal chain line is middle chainring in the front , and middle cog in the rear stack
# 3 of 5, #4 of 7, #5 of 9, etc.
they should be in the same plane , parallel to the centerline of the frame.

the 48 36 36 (or 24t) triple would be better with a Mountain front derailleur, but the current stuff for MTB/ are 44 t
may work .. their Trekking stuff is for bigger chainrings like a 48 but may not be in many dealers suppliers stock list.

If you were to compete in cyclocross the 46 30 is a decent choice the chainguard
plus a chain'minder on the inside should keep the chain from bouncing off on rough ground..

you have plenty of options

Option one: You can replace your current cassette with one that has a larger cogs, you can probably go as big as 30 (28 for sure). This relatively simple change would lower your lowest gear from 45 to 35 gear inches.

You could also replace your crankset with a compact crank, like the gran cru crank you linked to. A compact crank is a double crank that has smaller chainrings than a conventional crank (usually around 48/34). You can probably install that crank using your current front and rear derailer. That particular crank would give you a low gear of 30 gear inches with your current cassette, or 26 if you swap out your cassette as well. If you do this, your derailer might have trouble with the chain slack if you use your small/small combination, but as long as you're cognisant of this risk, you should be allright.

If you want true stump pulling gears, you're going to need a triple. This will require not only a new crankset, but also a new front and rear derailer. It would be an expensive endeavour, but you could get immensely low gears.

This is the part I don't understand. If you want lower gears, why not use the 39T chainring more? A 52/23 combination is the same as a 39/17 combination and gives you several lower gears beyond that.

I understand the need for lower gears for climbing but you should make better use of what you already have.

Because I can't climb the hills withe lowest gear the 39 offers, not even close. Even on the commute home with my panniers on with my laptop and other gear I could use lower gears on the one medium size hill that I have right before I get home.

For general riding, we have some big hills in CA. Many of the century type rides go over these hills. The 59-23 is when it is flat and no headwind.

My 'cross' bike has a 50,40, 24t triple and a 13-28t rear cluster, it works well .
low is 24/28 or 6/7ths = .8571428 :1 ..

if you got gear in the bags from the grocery run in the bags, and live on a hill,
a mountain bike drivetrain will shift OK with the bar end shifters..

many new touring bikes use that, or the road triple that adds that 30 to the 39 in the middle.