Really depends what you mean by 'catch up'. How much of a completionist are you? By and large, release order is a safe bet. If you're thinking purely in terms of the current, ongoing saga with the Mandalorian, I'd probably say something along the lines of

Original Trilogy (A New Hope, Empire Strikes Back, Return of the Jedi, films IV-VI)
Prequel Trilogy (Phantom Menace, Attack of the Clones, Revenge of the Sith, films I-III)
Star Wars: The Clone Wars (the CGI series, specifically)
The Mandalorian, seasons 1 and 2
Star Wars: Rebels
The Book of Boba Fett
The Mandalorian season 3
And then the upcoming Ahsoka series will probably be best watched next, though that isn't out yet.

You can kind of do Rebels and Mandalorian S2 in either order, Rebels comes first release-wise and chronologically but you're hopping around the timeline anyway, and I think Mando S2 serves as a pretty solid re-entry for the character the two series' share. I watched them in this order, and it piqued my interest as to what that character had been up to between here and Clone Wars, which Rebels answers.

rn I'm mainly watching Hajime no Ippo on my commutes since it dropped on my region's Netflix recently. At home I've got Samurai Jack, Our Flag Means Death and Hunter x Hunter on the go but not making huge amounts of progress through them.

Manga-wise, obvs there's Sket Dance, which I aim to finish at...some point in the next year lol. It's long for its genre and very dense text-wise. Also keeping up with One Piece, World Trigger and Chainsaw Man, but I like to keep my series' I follow chapter-to-chapter sparse.

Games-wise I've jumped from Breath of the Wild to Age of Calamity. I think I'm ready for another Warriors game to veg out to in a way I wasn't when it first came out. Also got loads of other games on the backburner (I mean, who hasn't lol) but the ones I seem to dip into the most are Celeste and Luigi's Mansion 3.

Capturing the narrative experience the player feels through gameplay without said gameplay present. Obviously there's no one-size fits all answer, but I think this can apply to a lot of them. Faithfulness alone isn't always enough.

As case studies, Danganronpa the Animation and the respective Persona 4 and 5 anime very faithfully recreate the plot of their games, but utterly fail to realise the experience of them. I remember watching DGR before playing the game and finding something really...off about how the cast investigate the various murders. There's just these extended, inert sequences of the characters spotting clues with no commentary on them, and then during the class trial Naegi just effortlessly strings all the clues together. There's a reason this rings so hollow- this is not how it feels to solve the mystery in Danganronpa games. As you're collecting each clue, you'll start speculating as to how they're relevent, and this deductive process continues throughout the trial. Compare this to any other detective story- there is always a strong sense of logical continuity. Detective Conan solves this problem by giving Conan a consistent interior monologue throughout the investigation of each case. If you're not involved in the deductive process, it just feels like watching a strategy guide.

Persona 5 the Animation bothered me for a different reason. I watched up to the end of Kamoshida's palace. For one, Amamiya is presented as a silent self-insert, rather than trying to embellish his personality based on Joker's more sassy dialogue options. The other is how...pointless the palaces feel. The dungeon-crawling experience is the challenge the Phantom Thieves take on. The arduous struggle of making your way through a complex labyrinth of enemies. And here they just get in and rush straight to the end to take on the boss, any exploration is skipped over. The plot of the game is stripped down to its cliffnotes and set-pieces, and as a result, nothing feels like it matters. Truth be told, I'm not sure an ideal anime adaptation of this could exist as things stand. The anime industry is moving to increasingly short-form productions and my blue sky pitch for a Persona adaptation would be a 50+ episode affair. For the palaces, particularly the early ones, I'd spent several episodes on the thieves getting to grips with the unique challenges each ones pose and the tougher fights they'd find with in. This isn't even getting into how you lace in the Confidants which...honestly, I'm not sure even translates mediums. idk how you present those in anime form without completely killing the pace dead in repetitive, intrusive ways. Anyway, I've gotten quite thoroughly away from the question at this point but I'll take any excuse to ramble, thanks.

Okay, least fave to best

Deep Breath (Capaldi) is the weakest to me from memory. I recall it just felt a bit...messy and weird? Wasn't really into the amount of comments Clara made about 12's age

Rose (Eccleston) comes up second last. I recall the episode being structurally fine and a decent enough way to reintroduce the series, but a bit lacklustre on the whole. The effects have aged...poorly.

An Unearthly Child (Hartnell) has a strong hook and establishes a lot of intrigue around 1, Susan and the TARDIS. Not the most substantial and leads into a pretty sluggish arc with the cavemen but a good opener!

The Christmas Invasion (Tennant) is pure mid-2000's BBC cheese and I love it. It's such a silly, fun way to introduce 10. The swordfight in a dressing gown and Harriet Jones' betrayal at the end are particular highlights

The Woman Who Fell to Earth (Whittaker) - Love the way this episode puts emphasis on the local community surrounding the new companions pitching in together to stop the alien threat. A really strong tone-setter that instantly sets 13's era apart from what came before. A darker, cinematic more grounded feel was exactly the change of tact the series needed (whether the rest followed through or not)

The Eleventh Hour (Smith) - There's a case to be made that this should be number 1. A really well-structured relaunch that delves into how an unfocused, reacclimating Doctor can effect his relationship with a companion through time travel. 11 is easily at his most likeable in season 5 and this ep does a great job getting you on board with him

The Power of the Daleks (Troughton) - it perhaps has an advantage being the only entry here with the benefit of a full serial's run...but damn this one is good. The first time the Daleks are really explored as a unique threat. The slow build-up of their machinations work in tandem with everyone's lack of trust in a new Doctor who's yet to prove himself. It's a perfect opening trial for 2's deceptively unnassuming schemer.

Looking forward to updating these when I've seen more Doctors!

Earthbound yes, Mother no. Earthbound/Mother 2 is pretty much a remake of Mother, with basically no direct callbacks to the events of the first game, and is much more accessible gameplay-wise. It also features the more distinctive atmophere and quirkiness that the series is known for.

Mother 3 on the other hand is a direct follow-up to 2 that iterates on what it does. Having seen lets plays of all 3 I'd planned to skip straight to it but now I feel that'd be a mistake. Earthbound is impeccably well-structured with a unique, delightful tone to it that should not be missed, and does a much better job of feeding you its mechanics in manageable chunks.

EB is a mostly cohesive journey where you acquire new party members and PSI abilities (spells, basically) at a steady rate. Mother 3, on the other hand, spends its first act jumping between player characters in service in service of a more narrative-driven game, before eventually giving you a full party to work with. But then once you have this full party, the game gives you new spells at a much faster rate to make up for lost time. As a result people familiar with Earthbound are much less likely to be overwhelmed.

It's beneficial from a story and mechanical standpoint to play Earthbound first.