I give you a lot of credit for trying to do the right thing. You don't need to do everything at once. Of course, in some ways (other than monatarily), doing it all at once is just easier.
Here's my suggestion:
Don't do anything until you have everything you need, some things may need to be ordered online if you can't find them locally.
Get a sump (30-40G would be ideal), overflow box (or two) and return pump, along with tubing for return pump. I'd go with AT LEAST 900GPH on the return. I really like Eshoppes overflow boxes. CPR makes a decent one as well, but personally, I ended up having far more issues with the CPR than I ever did with the Eshoppes. Get an overflow (or two) that can at minimum handle 200GPH more than the return pump you will be using (extra safe guard against flooding). As for the sump, if you're a Black Friday shopper and have a local *****, you might want to see if they're running any deals on their tanks. They run $1/Gallon sales on tanks every so often, and usually do on Black Friday, so you could save a lot and pick up a 30-55G tank for $30-$55 as opposed to what some sumps can cost. Measure under your stand ahead of time so you know what will fit.
Get a rubbermaid container, as large as you can find, drain the water from your tank into the container and place fish in the container with the Fluval running on it. Toss the rocks that you already own in with them, even the fake plants, so they don't feel completely out of sorts.
Either remove the crushed coral, OR mix it into the rest of the aragonite sand you already have. Add another bag or two of sand, it's really personal preference on how much sand since you're not going with a deep sand bed. I like to have a 2-3" sand bed personally. Here's the tricky part: Technically rocks should be added before sand. This provides a more solid foundation for the rocks, and then the sand acts to hold them into place. I have added rocks AFTER sand without any issues, but you really have to push the rocks into the sand and make sure they're secure. My suggestion would be to add in the rocks, THEN add the extra sand to add more support once the rocks are firmly in place. You can add in all the rock in one shot, or do a small order of half and half and then add to it. It's really up to what you want to spend up front, etc. Getting it all done at once really does make things easier. Try to get some branch pieces. Use these to create openings in the rock work, for flow and for your fish.
Set up overflows, sump, return pump. Add in your rock and sand. Fill tank with newly made water (put the water in by dumping it through a pasta strainer or splashing over a dinner plate, this helps to minimize how much sand it kicks up and the water will clear up faster. Let the new setup run over night while fish are in the rubbermaid. One night in the rubbermaid will not kill them, especially with the Fluval running. Definitely test, and if anything is out of whack, do a water change. For an added measure, you might want to pick up a small bottle of SeaChem Prime. Drop a cap full in when you first add the fish and water, it works as a stress coat AND can help keep ammonia levels down. USE ONLY A CAP FULL. Too much and it can de-oxagenate the water. If you end up having to do a water change, add another cap with the new water.
Next day: Test main tank. If everything looks good and the water is clear, toss in fish. Make sure you have extra water on hand as you may experience a spike for the first few days (week). Add fish with 1-2 cap fulls of Prime. Rinse the foam blocks from the Fluval in the left over water in the rubbermaid bucket that the fish were in and but them back in. Do a quick rinse of your media bags in the same water and toss them back in. Hook the Fluval back up to the 125. Since you own the Fluval, you might as well use it as extra water movement and a place for your media. I did this on my 155. Even though I had a full sump system setup, I removed all the foam and basically used the Fluval FX5 as a powerhead and a media reactor. After two weeks, remove a foam pad, then another, etc, all along testing.
Add CLEANERS.
This SHOULD help your issue.