There are tons of spells that increase survivability. Many of these will require a friendly spellcaster, but let's look at the ones a Ranger can cast:
- Goodberry: Heal your panther up to 10hp over 1 minute. For use between fights.
- Barkskin: Cast it on your panther to set their AC to 16 for the duration. (available at level 5)
- Stoneskin: Give your panther resistance to all nonmagical bludgeoning/piercing/slashing damage for 1 hour. (available at level 13)
If you're getting help from your party's spellcasters, you can also use Mage Armor (bumps AC up to 15), Bless (which will help on saving throws), and Aid (for temporary extra HP).
At higher levels, your party can also give it spells like Greater Invisibility, or Death Ward. Although it's worth noting, your party may want to use spells like these on their front-line fighters instead of giving them to your panther.
The bad news is, there is no "barding" or "animal armor" in the DMG, as far as I'm aware. The good news is, your panther can attune to magic items, and your DM gets to choose whether or not your panther can wear magic armor intended for humans:
Dungeon Master's Guide p.141: In most cases, a magic item that's meant to be worn can fit a creature regardless of size or build. Many magic garments are made to be easily adjustable, or they magically adjust themselves to the wearer. [...] When a nonhumanoid tries to wear an item, use your discretion as to whether the item functions as intended.
Items your panther would almost certainly be able to utilize include the Cloak of Displacement, Cloak of Protection, the Ioun stones of Protection & Mastery, or possibly even the Saddle of the Cavalier. If your DM is generous, any type of magical armor (Adamantine armor, Mithril armor, +1/+2/+3 armor) might also fit your panther.
Unfortunately, there aren't really any class features that specifically improve companion survivability. However, any class that has features that apply to their allies will generally apply to your panther as well. For example, your bard can give Bardic Inspiration to your panther. You could also take a level in Fighter or two levels in Paladin to unlock the Protection fighting style, which lets you use your reaction to impose disadvantage on an attack against your panther (as long as they are within 5 feet of you).
You'll probably want to keep the panther off the front lines, for the most part. At 50 ft per round, the panther is incredibly fast. Make good use of that by fighting in a hit-and-run style. Run up to a weaker archer or spellcaster in the back row, and Pounce on them. If the enemies start to close in, have it take the Disengage action (so it won't provoke attacks of opportunity) and run away. Note that this can be dangerous if you're unable to use your action to have it Disengage - provoking attacks of opportunity is a good way to take a lot of damage.
You can also make use of the panther's Climb speed to get to tricky places, or just to stay out of melee range. And while you cannot command it to use the Hide action, its +8 Stealth check may come in handy before combat begins.