Thanks sage advice. I have played through Morrowind, Oblivion, and Skyrim the first two without mods at all back when. For Morrowind it's more of a brush up to modern visual, widescreen and such like, and a bit of game speed up as movement is real slow. Otherwise I keep to minimum mods. Oblivion on the other hand had some terrible design decisions, the levelling of everything for instance really ruins the game, so I always used to play with Oscuros up until I found bevelixs mod list. I do need to go through again with that list. I really do think that out of all three oblivion is the one that really needs the most.

To be honest I am suprised that Beth's haven't remastered all the ES games into one package using Skyrim engine. One massive package would be great!

What mods you want are really just going to depend on your personal preference so I'll just answer your last two questions from your OP. I can't play Oblivion and have no useful opinions on it at all.

Tamriel Rebuilt is a lot of new area to explore. It's not as dense as the mainland but if you wanted more Morrowind in your game, then TR is a good choice.

Plain Morrowind is very stable. My pre-packaged install with all mods crashes maybe once every several hours in the later game when there's a whole bunch of stuff to keep track of (I hoard, in games). In the first 75% of the game it doesn't crash at all. So as far as stability goes, you're good with either one.

I'll weigh in for Morrowind mods:
I was looking for a stable game with fully compatible mods for mostly bug fixes, graphics and/or quality of life improvements. I'd love to take credit for researching/sourcing these, but they are more or less directly stolen from BTBs Morrowind mod list circa 8 or 10 years ago. His list has been defunct for awhile, but here's an archive from around the last time I downloaded all of 'em:

Edit: GOG ate my link, try this one in the wayback machine:
http://btb2.free.fr/mods.html

I'll just leave the list of the ones I use, go with his descriptions if you want to know more about individual ones. Also, while I only use a fraction of what he lists, it's still a longish list so strap in!

Morrowind Patch Project
Morrowind Cod Patch
Timeslip's exe optimizer (program for xp machines and the loose file for win 7)
Visual Pack
Bloodmoon Landscape Overhaul
Correct UV Rocks
Slartibartfast's Texture Fix
Connary's Texures
SWG's Skies (with BTB's fix)
Vurt's Hi-res Skies and Weather (excluding the night sky one)
Better Bodes
New Beast Bodes
Better Heads
Better Clothes
More Better Clothes (with it's patch)
Canadian Ice's Robes
Darknut's Weapon Textures
Darknut's Armor Textures
The three god replacers (Azura Replacer, Vivec God Replacement, and Better Almalexia)
Mesh Improvements
Darknut's Creature Textures (and it's patches)
Herbalism for Purists
Vality's Bitter Coast & Ascadian Isles Addons (with BTB's patches)
Morrowind Graphics Extender (3.8.0, I stopped looking before they made XE)

These all work together perfectly (if installed correctly and in the proper order) and make for a more stable than vanilla game to boot!

That is the list that worked for me, so I backed it up and stuck to it... well they didn't so much work for me when I started with them, but they do now. The Graphics Extender setup is a hog on resources, my 2014 computer is the first machine I've owned that can actually handle the game at the settings I wanted. Definitely worth it though! (glamour shot attached)

Is there an easy way to make Oblivion portable keeping all the enabled mods active? I got an old installation folder and a bunch of saves, but as I tend to build a new computer between the breaks I take on playing that game, I don't wish to figure out yet again which mods I need to activate unless I can backup the settings this time.

Best Skyrim mod: Bandolier - https://www.nexusmods.com/skyrim/mods/16438/

Not sure about portable, and depending on the mod sizes could be quite large. What I tend to do is keep all the mod installers, and from mod organiser or whatever you use, dump out mod orders and such like. Also I keep a text file log of what installs, when, what options to set etc.
The problem is really that in various mods you will need to set various options based on hardware, so a heavy mod you might drop down to medium res or something. I don't think there is one "build" of mods which runs everywhere and is portable, although several people have tried with massive mod pack installers (and modders really don't like them).

What you could do is store the folders as they sit on the drive, so your oblivion install. Then install oblivion on new system, and copy over the folders. Assuming you have all the relevant folders it *may* work. But personally I would always run through install from start to finish, hence the need for good solid build lists like Bevelix's
But isn't it what Elder Scrolls Online is? Take all the landmass, characters, places from all the games, drop them all into one, and not take over the rather vague small stories from each of the games? Hence it just means making an offline version, and applying the stories. Would be simple to my mind, I mean not Beamdog level of simple, would take a few man hours work, but not too difficult.
For those that just can't leave that last rat bone in the dungeon eh?

So I just want to add onto my original information that I have tried OpemMW. It is pretty good, plugged in a load of mods to it and it runs well. Will need to actually run through the whole game at some point, but it seems to be coming along nicely, with far less effort than MW+Overhaul.

Portability was just the shortest way to put in to words one of the problems I have with Oblivion, it just isn't nearly as mod friendly to someone who was used to making copies of Morrowind's installation folder and set different mixes of mods and texture packs on those without needing to use batch files to make any registry or folder name changes in order to achieve that.

I haven't played it online.If it was that easy then they would of done it and made the extra bucks.Also it would be DRM free and they could sell it on Gog.

In that case, I recommend looking through all mods by these authors. Pick what you like. They are all quality, and they are all wonderfully clever additions or changes to the game. Cheers!

Phitt - https://www.nexusmods.com/oblivion/users/128339
Miguick - https://www.nexusmods.com/oblivion/users/337763
JRoush - https://www.nexusmods.com/oblivion/users/453389
Tejon - https://www.nexusmods.com/oblivion/users/45898
TheNiceOne - https://www.nexusmods.com/oblivion/users/956375
Kyoma - https://www.nexusmods.com/oblivion/users/231999
ShadeMe - https://www.nexusmods.com/oblivion/users/644634
Arthmoor - https://www.nexusmods.com/oblivion/users/684492

I'll comment on openmw

I first discovered about openmw many years ago. Back when you can't even open a door without console command. Since then it grows into a very solid and stable Morrowind engine replacement. The goal of the dev is for Morrowind player to use it instead of vanilla executable and they do a good job so far. Openmw has been my goto engine for at least a year, whenever I feel the need to visit my 2nd home Vvardenfel

The current state of openmw is near finished. You can play the game from start to finish without any significant problem. The load is faster (scrawl, one of main programmer of openmw did some magical background autoloading thing), there's a basic distant land feature available (in vanilla you need me xe to have this feature), and much more stable than vanilla (my mgso modded installation crashed every half an hour on vanilla, while I haven't crash ten times in years using openmw). Openmw is available for windows, Linux, Mac, and Android. Yes you read me right. You can play Morrowind on your android phone using openmw although it's not part of the official version (one of the programmer decided to do it by himself and share the result)

Openmw however is not complete. Shadow is still missing. And pathfinding is atrocious, perhaps more than vanilla. Luckily right now those two big problems are being worked on with a promising result so far. There's also a problem with some textures being rendered weirdly and looks like a metallic object. This is because openmw uses "correct" way to render that's missing from vanilla (rocks replacer often faced this problem). There are probably a lot of things missing that I don't know about.

Regarding mods, openmw is in a rather weird position. The devs insist on not accommodating additional syntax introduced by Morrowind script extender (mwse) since it's obvious those additional functionalies can be added directly to the engine thus implemented much cleaner and with better results. This decision however, breaks many mods that depends on mwse. A lot of mods especially big sophisticated one like Julan is broken, and some modders have to create a "patch" only for openmw so you can play with Julan.

The other benefit of openmw is, you can actually make a standalone game using it. Some people are thinking about converting total conversion mods into standalone game. There was a talk about doing this to Ashes of the Apocalypse mod. But I haven't heard any news about it for long