I agree that something like an armored Cleric or Paladin would be a natural way of filling your party's Tank and Healer roles. But there are other options that could work too.
Druids can both tank and heal, as already mentioned.
And Bards can actually make good Healers in 5e. A Bard with 1 level of Life Cleric and a good Con score would make a fairly effective Tank and Healer, thanks to the Bard's spell list (which contains some healing and defensive spells) plus Life Cleric's bonus on healing spells and ability to use a Shield and wear Medium or Heavy Armor (depending on your stats). You could then decide if you want to focus more on healing by getting additional healing/restoration spells that aren't on the Bard's list by going Lore College Bard, or if you want to focus more on boosting your melee ability by going Valor or Swords College Bard. Being a Bard will also allow you to be the Party Face, with Charisma as your primary stat plus high skill checks for social skills.
There are also some other fun alternate options for a Healer/Tank that I haven't seen mentioned yet, like a Favored Soul Sorcerer (basically a Sorcerer with access to Cleric spells) splashed with a level or two of something like Paladin, Cleric, or Hexblade Warlock for greater melee/tank/front line ability. You'll definitely want to boost your Con to compensate for the diminished HP, though. For example, something like a Hexblade1/Favored SoulX has Medium armor, Shields, a few extra Warlock cantrips and 1st level spells, and nearly full Cleric/Sorcerer spellcasting, and only needs to focus on Charisma and Constitution (with a lesser focus on Dexterity up to 14) since one-handed melee attacks for a Hexblade are based on Charisma instead of Strength/Dexterity. You won't have the Extra Attacks of a dedicated melee fighter, but taking the scaling melee cantrip Booming Blade and/or Green Flame Blade will help your one melee attack to continue pumping out decent melee damage as you level up. You can focus your spellcasting on Sorcerer and Cleric defensive spells and Cleric healing/restoration/support spells, plus a smattering of other stuff like a couple blasty Sorcerer AOE spells. And being primarily Charisma-based means you can grab some social skill proficiencies and be an effective Face (although you won't have quite as many skills as a Bard).
A Levistus Tiefling would be a good race for either one of the above alternate Tank/Healer options (BardX/LifeCleric1 or FavoredSoulSorcererX/Hexblade1). It has a bonus to Charisma, a bonus to Constitution, Darkvision so you're not the odd man out, and some nifty additional racial spells, including Armor of Agathys for 10 extra temp HP 1/day. Human, Half Elf, or Scourge Aasimar also gets you a bonus to Charisma and Constitution. Hill Dwarf might be another good choice, depending on how you roll, due to the constant speed when wearing armor no matter your Strength score, the Constitution bonus, and the extra HP each level, although the Wisdom bonus doesn't help as much.
You could also try to talk the Rogue into going Thief subclass and taking the Healer feat too, which can make for some handy additional in-combat healing options. A Thief can use items like a Healer's Kit or Healing Potion as a bonus action during a round, and the Healer feat lets Healer's Kits not only stabilize downed allies but actually heal HP too to bring them back up to consciousness. This gets even a bit more effective when they have a high movement rate (from stuff like the Mobile feat and/or a race like Tabaxi), and can zip around the battlefield like an ambulance when needed.
