I started as a Ranger, and I'm almost to 3rd level. I am interested in multiclassing to add thematic and mechanical advantages, such as improved spellcasting and better armor proficiencies. Is there a class I can multiclass into which provides these things, and to which a Ranger foundation would be advantageous?
Rogue is a great multiclass for pretty much any base class, but (IMO) especially Ranger. It's a great thematic fit (both rogues and rangers are sneaky, agile, and tend to use finesse or ranged weapons), and you get a few incredibly strong benefits at very early levels:
Expertise gives you double proficiency in two skills, which is unbelievably strong. This will let you become (for example) an even better tracker, or extra super sneaky, or an absolute master of nature lore. Additionally, just for multiclassing you get one additional skill from the thief list (which is extensive).
You also unlock sneak attack, which is a hefty damage boost (an extra 1d6 per round is nothing to sneeze at).
Level 2 unlocks cunning action, which allows you to dash, disengage, or hide as a bonus action. This is a ridiculous buff to your action economy - being able to disengage or dash in while still spending your action to attack is extremely powerful. This also unlocks the dash-disengage combo, which allows you to cover double distance in one turn without provoking opportunity attacks. Hiding as a bonus action is also the key to a stealthy archer playstyle - you take a shot, move to cover, and Hide as a bonus action.
At level 3 your sneak attack damage increases to 2d6 and you unlock a roguish archetype. You could go Assassin, which gives you advantage on attacks against creatures that haven't taken a turn yet and automatic crits (!!) against surprised enemies. You could also go for Arcane Trickster, which gives you additional cantrips and spells from a different spell list, as well as additional spell slots (make sure to check out multiclassing spell slot rules in Ch 6). Arcane Tricksters also get a sexy upgraded mage hand that can do crazy things.
Improved spellcasting and armor proficiencies
Nature cleric is hands down the best choice of multiclass for these two areas for a Ranger.
You get heavy armor proficiency immediately at 1st level.
You get thematically sound new abilities (charm beasts and plants at 2nd level).
You get full spellcasting progression (including more nature-based spells) that matches the ranger's spellcasting ability.
And with ranger as the base class, you'll have access to the fighting styles to make you more threatening in actual combat, making you more effective as frontline support.
Cleric is the only thing that fits your requirements
Granted, not all domains will fit. Tempest, Nature, War, and Life all grant heavy armor proficiency, fulfilling your second requirement, and the cleric chassis is that of a full caster, accelerating your spell slot growth.
As a bonus, Ranger and Cleric both use wisdom as their spellcasting ability, meaning that this multiclass isn't going to make you MAD (multiple ability dependent).
Thematically, Tempest, Nature, and Life all fit with the ideals of a druid, but that's where your choice comes in, and I think it comes down to how you want to use your spellcasting feature the most. Tempest will make you a better blaster, able to cast more damaging spells; Nature will be an explorer, able to augment several of the Ranger's innate abilities with similar spells; and Life will be a healer who helps keep their team alive. If you choose to consider War, you'll be using your spellcasting to augment your martial attacks.