Hm. The new initiative change sounds like Numenera's initiative-- you succeed on the roll, you go before your opponents. If not, you go after. Maybe if the PCs roll initiative every round, it might make combat a little more hectic.
As for class-specific stuff, here's my two commons on the matter-- feel free to pick and choose whichever ones feel appropriate!
Fighters: Second Wind (immediately heal 1d10 + Fighter level HP, usable 1/day?), Weapon Specialization (every even level, +1 to attack rolls and +2 to damage rolls with one specific weapon, you the player choose, this can cap out at +3/+6 per weapon over the course of your career, or you can just say "I'm specializing in armor" and gain a Defense bonus 'cause I'll totally let you), Bravado (tag one enemy after a successful or failed melee attack roll, tagged opponent gets -2 on attack rolls that do not include you as a target, mark lasts until the end of your next turn and you only get one active mark at a time)
Thieves: Backstab (reach out and touch someone. In the back. Of the head. Does additional d6s of damage (or triple damage??), requires an ally to keep the unlucky bastard busy), Underdog (you gain a Defense bonus if the party is outnumbered, capping at +4(?) but this would make your Defense go up and down like a yo-yo as pals and baddies drop dead), Agility (you gain an additional bonus action that can only be spent on movement)
Clerics: Channel Divinity (deity-specific effects, along with Turn Undead and healing), Divine Smite (holding your deity's favored weapon? Enjoy better damage dice or additional fun effects!)... actually, I guess this is kinda like some kind of cleric/paladin hybrid?
Mages: Magic Missile (come on, this is a classic, EVERY wizard worth his spellbook should have this as a damn cantrip!! If that's a bit overpowered, there's always Eldritch Blast), Mage Staff (craft one of eight staffs based on the magic schools, gain effects while you're holding the staff, and you only get one staff, at least you can summon it at will), Metamagic (alter spells)
And for the love of Batman, please do for mage spellbooks what the D&D Next playtest promised (and I'm still irritated about that it's gone.)
And for character classes less commonly-seen beyond the Classic Four!
Alchemists: Alchemical grenades (boom! or brrr! or zaaap!), Chemical Resistance (resist poison and/or acid damage?)
Brawlers: Brawling (there's no finesse to it, you just beat people up), Dirty Fighting (open-palm face slaps, tripping a monster and kicking him while he's down, the Moe Howard Special, and the time-honored Kick-to-the-Balls no Jutsu!)
Psychics: Mind Thrust (melt brains! Make orc heads explode like in Scanners!) ...and other classic psychic powers such as pyrokinesis, cryokinesis, apportation, mind melding, psychic healing, astral projection, psychic empathy, telekinesis, and psychic blades (yes, the soulknife was my favorite classic psionic class, why do you ask?)