Can anyone recommend a desktop remote app that I can use to login to my Mac at home from my iPhone or iPad as well as login from my Mac to another Mac?
https://www.realvnc.com/products/ios/ may be a good option to try for the iPhone/iPad. I haven't used it, but have used the RealVNC products at work.
For logging in from one Mac to another, I use Screen Share from the Finder. Just click on the Mac you want to log into for the Remote Desktop session, and after authenticating; you're in. This assumes that your VPN/Firewall has the proper ports open on both networks.
I use LogMeIn. It just works without me having to do anything. Logging in from a Mac to another Mac might be a Pro feature now. I don't know. It's been years since I've done it because I prefer to login from my mobile devices.
If you’re on the same network, you can use Easy Remote Desktop. I’ve been using it for a couple of years. It does not, however, work if your devices are not on the same network.