Crusader Kings III

I recommend to start a new game with a custom character for this, since it doesn't take too much preparation and you will hardly ever encounter a situation in a normal game where going on such a journey makes sense (Except maybe in the late phase of a world conquest campaign when you want to do a really extensive grand tour or as a follow up to the "Far from home" achievement from the Northern Lords DLC).

The main challenge of covering all types of terrain is to get both jungle (which can only be found in India and Subsaharan Africa) and Taiga (which can only be found in the far north). It isn't possible to have the character just travel randomly, we need a destination that we plan to visit for a specific event and there is a limit on how much detour we can take along the way. The quickest way to get a reason to travel all the way from north to south or vice versa is to create a custom character either in the far south or far north and have them follow a religion that has a holy site at the other end of the world, so that we can do a pilgrimage to that place. You could e.g. create an Asatru character in the jungle of Africa, a Sikh in Norway, a Hindu in Siberia, a Tengrist in India etc.

To reduce travel risks, it is best to make a martial character with high martial and prowess and give him a few traits that have health boni (like e.g. stubborn, blade master) and the pilgrim trait for another bonus on travel security. Decent learning is also useful, so you can convert your council members, caravan master and court physician, since you'll probably have difficulties to find people to hire who already follow your religion. For the same reason, you don't want to have a big kingdom with many (hostile faith) vassals, because of the high rebellion risk. Best start as a small chief or duke in an isolated place and/or surrounded by weak neighbours and do a little bit of conquering and build up your troops to the point where none of your direct neighbours would consider you an easy target anymore before you set out on your journey.

If you start as tribal, the pilgrimage will be cheap and you should be able to quickly secure the necessary gold by doing a bit of raiding. To reduce the travel risks from the multitude of different terrains, make sure you hire a caravan master who contributes a high value to security (speed is less important), have a body guard and make sure your court physician joins you (If he has the learning education and is of your faith, that shouldn't be a problem. If he doesn't join, invest a little more gold for the next bigger entourage setting until he joins).

Use the terrain map and note some stops for the more rare terrains to make sure you really cover every single one of them. The terrain types are:

Taiga (only in the far north)
Forest (temperate zone)
Steppe (only in the temperate zone in Asia, different from Drylands!)
Wetlands
Mountains
Coastal Sea
(High) Sea
Oasis (surrounded by desert, same color as plains but the terrain type displays as "Oasis" when you hover over it)
Floodplains (Only around some big rivers in warm climate zones, different from wetlands!)
Desert
Plains
Farmlands
Drylands
Desert Mountains
Hills
Jungle (only in the far South of the map in Africa or India)

I got the achievement starting as an Asatru character in the chiefdom of Dibri (Southwestern coast of Africa) and conquered the duchy of Kru before I made my pilgrimage to Uppsala (with a little detour to eastern Europe to cover Steppe). Despite the huge distance, the pilgrimage cost me less than 300 gold on the second-highest entourage setting. I won't list the specific holdings I used as stops in between to cover all terrain types, since there are many possibilities, just use the terrain map.

Solution taken from the CK3 wiki, with additional geographic guidance from me.

Start as a custom character as the Chiefdom of Obran Osh in 867 with Vaishnavism (Hindu) as your faith and Pilgrim as a trait. Put your remaining points in whatever you feel like. You can immediately perform a pilgrimage to Ujjayini. Click on customize route in the bottom left hand corner. In the window that pops up click on the + symbol to add detours.

Program your route as:
Obran Osh - Start (North of Khazaria)
Theodosia (Crimean peninsula)
Alexandria (Egypt)
Ujjayini - Destination (India)
Rajpipla (India- north west end of the Rashtrakuta Kingdom)
Baqirda (this was the hardest to find, it is in the Wilayah of Hakkari, roughly north central region of the Abbasid Empire)

This route will hit the 14 terrains on land and 2 at sea, at least once at:

Obran Osh - Taiga
Mtensk - Forest
Chally-Kala - Steppe
Or Qapi - Wetlands
Theodosia - Mountains
Black Sea - Coastal Sea
Levantine Sea - Sea
Alexandria - Oasis
Cairo - Floodplains
Quzlam - Desert
Kamboika - Plains
Ujjayini - Farmlands
Rajpipla - Jungle
Basiyan - Drylands
Baqirda - Desert Mountains
Saird - Hills

Achievement should register before you make it all the way home.