Recommended focus for a new player?

AP limits everyone's gameplay. If you really want to be efficient than yes you should clear as much of the shop as you can. The shops is definitely the most ap efficient way to get exp tokens as well as the best guaranteed way to get ascension and skill up materials.
If you want to truly enjoy the game though, keep at it with the story. Not only is it more content to clear but also it increases the pool of servants you can get as SRs and SSRs.

My personal suggestion: at least buy the exp tokens and maybe even the fous. And buy whatever ascension tokens kiyo and any of your other high use servants use if they're available in the shop. That way it becomes a choice after whether you want to keep efficiently farming the event or want to progress the story (which by the way would get you more quartz to summon so is also beneficial). The most efficient time to do the story by the way is when it can be done at half AP cost which does happen in between events.

The story mode is not going away, like the events. You can allways do it later (Hint: Orleans, the chapter after Fuyuki is full of Wyvern, who are Riders; bring your best assassins). But they do liberate extra servants in the gacha poll...

Still, I recomend to do Main Plot when there is no event going on (or if the Event is Nero Fest, who is just so hard for newcomers). Extra bonus if it is a "half-AP Main Quest campaign", where you can rush the first chapters in like 2 days. Most events do only demand Fuyuki to be done, exceptions for some (Halloweens demand Orleans completion, for example).

Two recomendations: 1- Don't waste your apples for more Ember/Materials quest (or in the main story), they are rare, keep for events.
2- Have in the begining one (or at maximum 2) main Archer, a main Zerk, a main Caster, etc., and throw the maximum of class-related embers you have at them. After they reach a certain level (let's say, 40), you can start working on another unit. I recomend (just silver and bronze ones, as they are the ones who you are very common to roll) Cu/Proto-Cu for Lancer, Robin/Euryale for Archer, Hans/Medea for Caster, Caesar for Saber, Cursed Arm Hassan or Kotarou/Kojirou for Assassin, Kiyo/Lu Bu for Berserker, Ushiwakamaru/Medusa for Rider.
Oh, congrats on your Kiyolancer, as she is a excellent Buster and boss killer.

As for mats, for now priorize second ascensions of the servants over skills (well, they only start to use ascension mats by level 4-to-5, so you can keep them at 4 in the begining).

Also, roll your Free 10X FP gacha everyday, most of the content is trash (1* CEs, feed them to your good CEs), but you can occasionally get good 3*s servants and CEs, and even NP up/limit break them (also, NP5 Hans, Kojirou, Georgius and Arash, who are very useful low star units).

Hmm, since you've only cleared Fuyuki at this point, might try doing a story quest until you level up. Then do maybe one or two quests at the summer event then save some AP enough to go for a few story quests then use that AP again to level up at story missions. APs were never a problem at the early stages of the game iirc. I just kept doing story quests and used what meager drops I got to level my servants and let supports carry me through the story. I never ran out until I think Okeanos or Rome.

Going to add a bit more detail on the non SR servants to help based on what you have:

Saber: Caesar is your best option for now. You may get spooked by OG Seiba (Altria) or Daddy Issues (Mordred) both of whom are good as SSRs but the odds of that are low so Caesar has a good kit and Ugly Bastard is definitely worth levelling.

Lancer: Kiyo is honestly really good so you're somewhat set there. If you want a backup who can potentially even be used in end game level FSN Cu (Fate stay night or just normal Cu Chulain). Protection from arrows is an amazing talent that makes him near unkillable.

Archer: Robin hood is a very good boss nuke but poor at everything else. I'd level Euryale first as she is about the same as Robin while also having the amazing niche of being amazing against male servants. She can stun them quite a bit with her NP and her second ability and does do more damage against males.

Rider: Ushi is quite decent. Christmas event will bring out Santa Alter I believe as the welfare servant who is quite a decent rider herself

Caster: Hans Hans and Hans again. Get a Hans, np 5 that Hans, level that Hans. Outside of waver or Merlin (both SSRs) there is no better support unit. If you want to level another caster, Medea has a unique ability to cast her NP often and remove boss buffs with her NP which comes in more handy than you'd think!

Assassin: Shishou would be your best option as she is so easy to ascend. If you get the ascension materials though, Fouma is the second best star generator after SSR Jack the ripper.

Berserker: Bahsahcar is the best option if you can ever get him on rate up or have him spook you (I mean Heracles). But failing that, Lu Bu is a very decent berserker that puts out quite a bit of buster damage as long as you can keep him alive. Special mention goes to Kiyohime who is almost worth raising for her voice interactions alone. Kiyo stars primarily as a good farm servant who can AOE np sets of mobs down.

Other: Level best kouhai. She costs 0 and will never give you up, never let you down, and even never run around and desert you.