Reroll targets

Following list contains unit recommendations broadly agreed on by the community, intended to help players starting out.

Keep in mind that due to variety of game modes and damage types there can be no universally applicable tier list, so the suggestions listed are a matter of opinion and are meant to be a starting point rather than a definitive guide.

Who to choose?

Recommended students who are basically always good
Iori	Iori has very high damage output at low EX cost. Performs extremely well in PVP and in Raids against Heavy armor bosses. She eventually becomes farmable in Hard 14-3, but having her early on is a very significant headstart. Useful in all content, really, and arguably still the strongest unit in the game
Ako	Ako won't help you too much when you're starting out, however she becomes a must-have for anything outside of PvP eventually. Ako is a buff support character who increases characters' crit damage stat. Crit damage has, since launch, proven to be a king stat. She targets one character for a fairly low cost, and makes them exceptionally powerful. Also, she heals a party member every half minute or so, which really comes in handy. Not farmable.
Cherino	Cherino has a really useful ex skill (a humongous yellow screen clear) that gets more useful when maxed out (it gets cheaper), but more importantly she has a so-far unique ability to increase the "cost" rate regeneration. This means more ex skills, very quickly. She has yet to stop seeing use. Not farmable, not even really from that one event where you can buy her shards.
Strong at the start, but later gets frequently replaced
Hibiki	Hibiki is an extremely versatile Special (off-field) unit that sees play in all game modes. Has damage output on par with striker characters. Remains irreplaceable in PVP, simply for how wide and powerful her skill is. Not farmable. The only issue with Hibiki is that Ako has sort of replaced her. As Ako makes strikers into hypercarry units and provides the same default buff as Hibiki, Hibiki has often been shelved. This has gotten even more ridiculous with the introduction of Hoshino (Swimsuit), who also buffs strikers and in a completely broken way. Hibiki has been down, but not out, and regardless she's very useful when starting.
Always pretty good
Shun	Shun provides skill cost boost at the beginning of combat. Core PVP unit; sees some play in Raids and Normal/Hard/Challenge missions for extra fast clears or as a general purpose Explosive damage dealer. Not farmable.
Koharu	Koharu is one of the few Healer role Strikers, arguably still the best one, and using her allows you to free up a Special slot that would normally be taken by a healer, or to bring more healing to any battle. Healing grows in importance for Extreme+ difficulty raids in general, Hieronymus especially has stage mechanics centered around healing. Not farmable.
Nodoka (Hot Spring)	Nodoka (Hot Spring) is a healer who's particularly great at healing. In fact, she ends up being quite the savior in a lot of difficult content as well as in pvp. Coupled with a passive attack buff, she's great to have around. Not farmable.
Good for their color of content
Aru	Aru is an Explosive damage unit who eventually becomes so powerful that she's meta. At the start, she's a solid if somewhat unwieldy pick. She has an AoE/single target ex skill which can either hit a lot of people or just one. Farmable, but only farmable late (from Hard 18-3). Seeing as she's not of critical usefulness at the start of the game, it doesn't exactly hurt to wait and not reroll for her..
Arisu	Arisu is a Mystic damage dealer who can deal RIDICULOUS amounts of damage, but at some cost. Her ex is expensive for one, and it's best used after waiting for her to charge up twice. Her cost can be reduced by Ui and she can be paired with Karin (Bunny Girl) for positively devastating results if you have Ui, but these are often niche strategies. As she is not farmable, getting her early is pretty great.
Iroha	Iroha is an extremely powerful Mystic damage specialist with a somewhat overblown and subsequently undersold reputation. Iroha is not farmable and is exceptionally strong. However, Iroha's most useful content is against the raid Perorodzilla and within pvp. Many other units are used before her in a lot of content, but that doesn't mean she's bad. Be happy if you roll her.
Saki	Saki is a damage-dealing piercing specialist particularly useful against the raid Hod. She's also notable for providing just about the best attack boost you can get from specialists. She isn't HIGHLY useful, hence her being at the low end of this list. Karin provides similar effects and a more powerful ex skill, however she is actually quite easy to obtain for a number of reasons, including being farmable (albeit late). Saki is not farmable.
Haruna	Haruna is a Mystic damage counterpart to Iori. Just like Iori, her EX can devastate enemy teams in PVP; solid damage dealer in blue raids. She is easier to farm (hard stages 8-3 and 12-3) than Iori, which makes getting Haruna from gacha less of a priority.
Good, but niche
Ui	Ui is a non-farmable, extremely unique unit. Kokona can provide something similar but not too reliably. Ui can cut the cost of a single strikers' ex skills by half (rounded up for odd numbered costs) for 2 uses of that skill. Obviously, that's awesome. Unfortunately Ui is a striker herself, and double unfortunately other strategies simply tend to be more effective. Ui dies very quickly if she's targeted, even at high levels. Still, a nice option to have.
Saya (Casual)	Saya (Casual) is basically "the same" as Hibiki. They have similar stats, the same passive buff, and strong skills. Casual Saya, however, costs more and is generally nowhere near as effective as Hibiki in general. Still, she's pretty good, and nice if you didn't get Hibiki.
Notable mentions
Tsubaki	Tsubaki is the premier tank, despite being 2-star. Extremely hard to deal with in PVP early on, remains a frequent PVP unit as of near 2nd anniversary. Isn't farmable until later (Hard 15-2), but given rarity you are likely to get her during initial pulls. Has a taunt that is extremely useful.

Who else can you use?

Following units are either farmable through various means, or relatively easy to obtain due to lower starting rarity.

Serika	Serika the 120-yen cat can be bought outright with a starter package or easily farmed. A well-rounded damage dealer, high investment Serika sees play in quite a lot of Red content, though there are quite a lot of Red girls nowadays who can make her somewhat obsolete.
Serina	Serina is your primary healer. Extremely versatile and easy to use, she provides excellent single-target healing, 2-cost EX skill can also be used to reposition units. Does not need high investment early on, but becomes increasingly important in later raid content. Core 1st season PVP unit.
Kotama	Kotama provides valuable Attack buff for your Strikers. Sees a lot of play in raids.
Maki	Maki, obtainable from the raid shop. Provides Defense shred and solid single-target damage output for Binah raid.
Akane	Akane, obtainable from the raid shop, is another source of Defense shred. Does same Penetration damage as Maki, but trades damage output for a higher Defense reduction with better debuff uptime. Akane and Maki can see some play on other raids early on, but with Azusa and the limited Azusa (Swimsuit) providing similar Defense reduction effects in other damage types, their niche is primarily Binah raid.
Nonomi	Nonomi is a free unit provided through starter campaign. Widely used for story content, clearing EXP Commissions, and sees play in Chesed raid. In a pinch, she can be used in PVP early on if you are missing better damage dealers.
Fuuka	Fuuka fragmets are sold in Arena shop, she provides same healing and repositioning mechanic as Serina, but for up to 4 characters. Useful as second team healer for clearing story content, core for Extreme difficulty ShiroKuro raid due to repositioning mechanic.

Note that characters sold in Raid and Arena stores are relatively easy to max out in a few months of play.