I'm looking for a way to have an Animal Companion with classes other than Druid or Ranger, for example a Paladin that fights with a trained hawk. I'm all for multiclassing if need be, but I'm looking for other ways as well, through feats if necessary. I'm not really worried about min/max stats or optimized builds.
[RPG] have an Animal Companion without being a druid or ranger
If using Pathfinder, you can use the Summoner class from the Advanced Player's Guide. Though it's a caster class, it's got d8 hp, and can do "no armor, spear, knife" just fine. Have your eidolon (powerful pet) be a tiger, and then your other Summon Monster uses can be Tarzan-like bellowing to the jungle. Works from level 1 and scales. This means the tiger will take the lions' share (liger's share?) of your combat activity.
As a druid, you could just have an animal that uses the Cat, Small stats until 4th level and then grows into the Cat, Large stats - just instead of swapping out companions, say it's the same one. Ta da.
Take an appropriate class template like the Jungle druid, or Beastmaster ranger, and then take the Boon Companion feat which ups your effective level by 4 for animal companion purposes.
Posted originally in 2008 on the Wizards of the Coast boards here and updated in May 2010 on the Minmax Boards here, Surreal's List of Stuff, in addition to providing a great deal more, lists classes that improve one's animal companion. I assume--given the amount of research in other areas--that its animal companion information is comprehensive (the absence of the halfling outrider notwithstanding), but it does exclude Dragon and Dungeon magazines.
Below I've confirmed that the prestige classes do, in fact, advance animal companions as the List of Stuff's commentary is either absent or telegraphic.
From the List of Stuff
The prestige class abolisher (LoM 182) at level 1 gains the ability animal companions. It reads, in part,
If you have an animal companion, your abolisher levels stack with your previous class levels (usually druid or ranger) to determine the animal companion’s abilities. (183)
The prestige class arcane hierophant (RW 108) at level 1 gains the ability companion familiar. It reads, in part,
You add your arcane hierophant class level to your druid or ranger level for purposes of determining your animal companion’s bonus Hit Dice, natural armor adjustment, and Strength/Dexterity adjustment (see the sidebar The Druid’s Animal Companion, page 36 of the Player’s Handbook). [...] You add your arcane hierophant class level to your arcane spellcasting class level, and determine the Intelligence bonus and special abilities of your animal companion accordingly (see the sidebar Familiars, page 53 of the Player’s Handbook).
The prestige class animal lord (CAd 22) at level 1 gains the extraordinary ability animal bond. It reads, in part,
If an animal lord has the animal companion class feature, he adds his animal lord level to his effective druid level for the purpose of determining his animal companion’s bonus Hit Dice, special abilities, and so forth. (23)
The prestige class beastmaster (CAd 26) at level 1 gains the extraordinary ability animal companion. It reads, in part,
If a beastmaster already has an animal companion from another class, her beastmaster class levels stack with class levels from all other classes that grant an animal companion.
The prestige class gatekeeper mystagogue (PE 87) at level 1 gains the ability animal companion. It reads, in part,
If you have an animal companion, your gatekeeper mystagogue levels stack with your previous class levels (usually druid or ranger) to determine the animal companion's abilities. (88)
The prestige class lion of Talisid (BE 65) at level 1 gains the extraordinary ability animal companion. It reads
A lion of Talisid can add his lion of Talisid levels to his effective druid level (his actual druid level or one-half his ranger level) to determine the capabilities of his animal companion. (66)
The prestige class peregrine runner (RS 116) at level 1 gains the extraordinary ability peregrine companion. It reads, in part,
If a character already has a peregrine falcon [that uses the statistics for a hawk (MM 273)] animal companion from another class, the levels of that class stack with his peregrine runner class levels to determine the falcon’s power and abilities according to the animal companion table on page 36 of the Player’s Handbook. (118)
The prestige class planar shepherd (FE 105) at level 1 gains the ability animal companion. It reads
Your planar shepherd levels stack with your druid levels for the purpose of determining the abilities of your animal companion. (106).
The prestige class revenant blade (PE 142) at level 1 gains the ability ranger class features. This reads, in part,
If you have ranger levels, your revenant blade levels stack for purposes of determining the abilities of your animal companion. (143)
The prestige class Vandalis beastkeeper (Dra 130) at level 1 gains the extraordinary ability improved companion or mount. It reads, in part,
Levels in Vandalis beastkeeper stack with levels in the class that grants you your animal companion or special mount for the purposes of determining the functional strength and level of that companion or mount. (131)
The prestige class wavekeeper (Sto 76) at level 2 gains the extraordinary ability beast of the sea. It reads, in part,
Starting at 2nd level, your animal companion continues to advance. Subtract 1 from your class level and add the result to your effective druid level for determining the abilities of your animal companion. (77)
The prestige class wild plains outrider (CAd 92) at level 1 gains the ability animal companion/special mount. It reads, in part,
A wild plains outrider adds his outrider class levels to his effective druid level (his actual druid level or one-half his ranger level) to determine the capabilities of his animal companion. (93)
Not from the List of Stuff
The prestige class halfling outrider (CW 38) at level 1 gains the ability mount. It reads, in part,
Halfling outrider class levels stack with paladin, druid, and ranger levels for determining the characteristics of a paladin’s mount or an animal companion. (39)
The prestige class legacy champion (WoL 19) at every level except 1 and 7 gains the ability class features (seriously, the special ability is named class features so it has a class feature called class features). It reads, in part,
At each level except 1st and 7th, you gain class features and an increase in effective level as if you had also gained a level in a class to which you belonged before adding the prestige class level. The specific class features you gain include spells per day (and spells known, if applicable), improved chance of turning or destroying undead, metamagic or item creation feats, bonus feats, monk special abilities, sneak attack progressions, and so on, depending on the class. [Emphasis mine.] (19-20).
Ten levels of the class should increase by 8 the character's level for his animal companion ability among all but the most conservative DMs but, as always, ask.
I'm happy to update this based on Comments and my research. As the question asks only for prestige classes, just those'll be listed, despite other animal companion resources being awesome (e.g. the feat Devoted Tracker (CAd 108), the alternative class feature urban companion ("Cityscape, Part 1: Urban Class Features" here).
There is a plethora of classes that either have an Animal Companion as a class feature by default, or can gain one via Archetypes.
In particular, the Huntmaster Cavalier comes pretty close to your idea of a Paladin with a Hawk, albeit without the inherent holyness that a Paladin has.
Alternatively, if you want to keep the clerical aspects of the Paladin, rather than the knightly ones, a Sacred Huntsmaster Inquisitor might be worth looking into.
The following list is limited to Paizo content and not exhaustive, particularly towards the end (I didn't actively look for Archetypes that grant a mount, for example)
Standard classes with Animal Companion
Druid, the iconic AC class.
Rangers can get an AC via Hunter's Bond, at lvl-3.
The Hunter from the ACG is a Druid-Ranger-Hybrid, with a uniquely strong focus on the animal companion.
Cleric, via Animal Domain (or it's subdomains Feather and Fur), at lvl-3
Classes with Archetypes or Options that grant Animal Companions
Barbarian, Mad Dog: Gains Improved Drag and can use it as swift action to pull an enemy toward his master.
Ranger, Falconer: bird only, full effective level, can learn special trick.
Alchemist, Winged Marauder: Dire Bat or Giant Vulture only
Cavalier, Huntmaster: Bird or Dog companion, can benefit from the Cavalier's Tactician ability, gains extra effect on it's attack.
Inqusitor, Sacred Huntsmaster: gains master's Teamwork feats automatically.
Oracle, Lunar Mystery, Primal Companion revelation: bear, boar, crocodile, shark, tiger, or wolf.
Brawler, Wild Child: learns the same Combat maneuvers as its master.
Classes and Archetypes with features that "function as a druid's animal companion"
Paladin: Divine Bond can be a mount.
Cavalier: Mount class feature.
Bloodrager, Bloodrider: gains a Feral Mount at lvl-4.
Shaman of Nature or Mammoth Spirit can have thier familiar gain the physical aspects of an animal companion - albeit at level 16, BUT, with the bonus of still being able to use familiar abilities. Mammoth spirit does have to be a megafauna companion, though.