Absolutely possible. I know several people who maintain their characters like this - playing from an early level, working for the most part as a unified group. It's technically possible to play a party of up to 4 people this way - but that has its own scheduling challenges to contend with IRL.
Main story is "solo" to an extent - cutscenes will (with a few exceptions) be purely your character. You'll be playing the same story* (one note below) at the same time, and hit the same dungeons at the same time. And it's a good story - it starts off kinda slow to be honest, but it gets going if you stick with it. Some of the best story is in the first expansion and beyond - which is accessible to free trial (effectively free to play) players (the base game and first expansion that is - soon the second expansion will be added to the free trial).
A quick aside so you're aware - it is a largely linear JRPG story. I'm not sure what your goal in a video game is, but if endgame raiding is your goal, there's gonna be some story to get through. Even the raids themselves have stories attached to them as you play through the tiers. IMHO it's worth it to watch/read the stories, but if skipping is your mantra, there is a cutscene skip button that works most of the time (some cutscenes cannot be skipped)
Out in open world/field areas, your party will start to matter. Here, you can fight with your party members at your side, and mobs you kill will generally grant all party members equal experience, rather than having to split it up. (There is a bit of math to when you gain exp based on party level, but that's beyond the scope of an intro post)
As you progress the story, you will also encounter "solo instances" of a specific main story plot-related boss fight or event. You will need to drop your party members to complete this (while they complete it in parallel if they are at the same story quest). You can reform your party manually when both finish.
Dungeons are generally always multiplayer. They will be 4 players comprised of 1 tank, 1 healer, and 2 DPS. If you enter the dungeon without a full party makeup, the game will automatically matchmake you a party from a pool of people who registered for either that dungeon or various "randomizer" queues. When you enter, your level will be synchronized down if you are too high a level, so you don't have to worry about high level players flattening things while you do it for the first time (for the most part, gear doesn't always scale down well). Once you unlock a dungeon, you can just register for it with your party as many times as you want - but sometimes the main story will dictate that you complete it at least once.
Trials are larger scale boss fights comprised of 4 or 8 players depending on the trial. In the 8-player format, the party composition will be 2 tank, 2 healer, 4 DPS. Like dungeons, these are matchmade if you do not enter with a full party. These will take place at critical points in the main story, and are associated with some of the "bigger bads" of the story (no spoilers shhhh). Most trials will come with an extreme version - more on that later.
Raids follow the same party composition as trials, and are optional parts of the game. The quest to unlock a raid will be able to be accepted after you complete the main story of the base game or a given expansion. With the exception of the first raid series (for the base game), all raids in this format have a normal difficulty and a savage difficulty. Like Trials, if you do not enter with a full party, one will be matchmade for you.
Savage and extreme would be the "endgame" raiding of a given expansion. It has the same 8-person party format and much tougher game mechanics to learn. Although it is level synced by default, gear levelling tends to make older content easier as you get further past that in the main story.
There are also 24-person alliance raids comprised of 3 parties, each with 1 tank, 2 healer, 5 DPS. There are somewhat of a hybrid between the dungeon and the raid - higher end mechanics than the dungeon, and a little harder than a dungeon with an equivalent level. Like dungeons, trials, and raids, these are matchmade.
When you start the game, your starting city will be assigned by your class. You'll need to play the story through to level 15 (iirc) before you can travel to the other starting cities. If you want your friend here with you while starting out cold, you may want to select classes with the same starting city. Don't fret if your favorite classes aren't in the same starting city - the first multiplayer dungeon is AFTER the quest that lets you get to the other starting cities.
Once you reach a certain level (10 if I recall correctly) you can pick up additional classes. Levels are assigned per class, so you can always change up your whole kit without re-rolling from character select. Your class at time of combat start is determined by which weapon you're holding in your hand - be it a gladiator's sword, a conjurer's staff, or an arcanist's tome. The game however will prevent you from changing classes in the middle of combat :P
Personally, if you're starting out as a duo, I'd recommend classes that do not overlap in a dungeon - for example, you won't be able to use the matchmaking system if you try to enter a standard dungeon as two tanks or two healers, but you can enter an 8-person trial or raid with that same composition. You can however pick the same DPS class and pair up in dungeons since there are two DPS positions.
Oh and if you find out you have friends in game but you picked the wrong server? Cross server travel is accessible without logging out from the three starter cities, and cross-datacenter within the same region is accessible from the login screen.
Most importantly, if something is confusing, ask away! The vast majority of the community in my experience (over 6k hours) is pretty chill and willing to help and/or answer questions. I also realized I just dropped a wall of text on you to a simple question... Seriously ask if something is odd sounding.
May your journeys fall in the light of the crystals!
-- Ella Wis (Coeurl - Crystal datacenter)