Unless specified otherwise, you need to defeat the monster in battle before it will join you. The story monsters are listed for completeness. You will be forced to get them as you progress through the game.
It is a good idea to get each secret monster as soon as it's available, so that their levels won't lag behind the rest. You should then be able to get everyone to level 30 before departing for Chapter 11.

• Leviathan - Chapter 1. Dr Blood's starting monster
  
• Mr. Beaks - Chapter 1. Gabe's starting monster
  
• Philosofly - Chapter 2. Jim's starting monster
  
• Brodent - Chapter 2. Moira's starting monster
  
• Dust Bunny - Chapter 2. Brahe Castle.
  
• Tycho - Several places. Joins automatically, but usually leaves soon.
  
• Goomez - Chapter 3. Monster train.
  
• Vendorr - Chapter 3. Monster train.
  
• Twisp - Chapter 4. Desert, immediately outside Cirroca.
  
• Catsby - Chapter 4. Desert, immediately outside Cirroca.
  
• Fuchsia - Chapter 6. Hestia's starting monster.
  
• Grouchophon - Chapter 6. The Jungle.
  
• Infinitoad - Chapter 6. The Jungle. (End of the second section.) Battle won instantly.
  
• Murch - Chapter 7. Yggdrasil. (First ascent.)

• Scream Cone - Chapter 1. Ice Cream Cave. (West of train station.) Defeat the boss, then walk back. If you are having trouble getting through the cave, there is a useful Molotov item in the first room of the train. Just do NOT pull the switch there, or else the train will start moving and you won't be able to get Scream Cone until Chapter 10.

• Coral Golem - Chapter 8. Coral Cave. (Pictured. From Effluvia, go a bit south, a bit west, then due north.) Go up behind the waterfall. (Pictured.)

• Broodax - Chapter 8. The Battleground. (Pictured. Third segment, just above the bridge. In plain sight, but can be missed if you go left instead.)

• Ninja Z - Chapter 9. Q'atra. Third floor. Go through the red portal, then move up behind the pillar. (Pictured.) Note: since Q'atra will collapse afterward, you can only get Ninja Z during chapter 9!

• Nigiri-Chu - Chapter 10. Effluvia Arena. (Pictured.) Defeat Fish Force.

• Cthulhu - Chapter 10. R'lyeh. (Pictured. Southwest of Brahe Castle.)
  
• Dr. Missingno - Chapter 10. A trainer. Defeat him in the northwest corner of the map.

Scream Cone
Scream Cone will probably not be a heavy hitter, because his powers are divided between strength-type and magic-type and aqua is not usually super-effective. However, he shines in a selfless support role. Flavor Wave (damage Scream Cone, heal rest of party, 1MP) is great if you can occasionally spare a potion/healing power heal from another party member. Frozen Treat (transfer all MP to another party member) is pretty good if done regularly with Grouchophon and potentially great if timed well. Blizzard and Icicle have their uses occasionally.

Coral Golem
Initially he's a tank, and first becomes available at a time when you can really use one. Shieldmaster is nice too, but his real strengths come later. At lvl22 he gets Waves of Healing (Passive, regenerate party members while Golem is alive), which lets you use him instead of a dedicated healer in the easier battles at least.
At lvl30 he gets Apocalypse, one of the best attacks in the game. It's a heavy magic attack all which is almost always super-effective, regularly dealing 7000+ damage. Add Talismints and Moira's +1MP ability and you can use it in turn one! To maximize damage I recommend pairing Coral Golem with Jim until he hits lvl30.

Broodax
Another tank. You'll mostly want to alternate Scarapace (Medium attack, dam based on HP) with defending (regain HP through armormaster). His ability to suddenly resurrect with a lot of HP is quite nice. Consider your trainer choice carefully: Jim gives +1MP each time you Defend, but Moira gives more appropriate stat growths (as well as +1MP at start).
Would be a good candidate for the Energite item (+1MP/turn) later on except that that's still better in the hands of a magic user.

Ninja Z
Insane speed, making Ano Eto Sa! (light attack, super-effective if faster than enemy) rather good. Pretty good for using items quickly. But like Coral Golem, Ninja Z's hidden strength comes at lvl30. At that time, he gets Sekai Sen, a passive ability adding Holy to all his attacks.
Holy is super-effective against almost everything in Chapter 11. Using Moira, you can use Shinde Kudasaimasen (medium attack all, 2MP) in turn one. That means between him and Coral Golem you can end most Chapter 11 battles before they get started. The first chest in Chapter 11 contains a Wafer, an accessory that heavily boosts Holy damage. Make sure to equip that.

Nigiri-Chu
An Aqua-based magic user, starting out with one the highest Magic attributes around. His Stench ability (1MP reduce resistances and defenses on all enemies) is extremely useful, and Drown routinely does well over 10000 damage.

Cthulhu
Joins at lvl30 with intensely high attributes. Powers are decent and straightforward enough. Also all physical, which is nice. His special ability does 500 damage all each turn without doing anything, but also 25 damage to each ally. That's often a good deal if you have ongoing healing, but not always.

Dr. Missingno
A top-level Dr. Blood. That's pretty much it. The only ability he has that Blood didn't have when you lost him is Glitch, a heavy dark magic all for 4MP. Not bad, but not better than the River Styx power Leviathan gets at lvl30 and generally not good enough to justify using him over other trainers. His Dark-boosting power means you could consider using him for Goomez, but that's about it.