How to Raise a Powerful Monster in Monster Rancher 3
Before you start following this guide you'll need to purchase the Holy Goblet, Beaming Rock and Aroma Pot. These three items can be purchased for extremely high prices but they grant you passive benefits which allow you to train your monster for much longer periods of time without needing rest. If you can't afford these items I recommend you level a monster up, freeze it and unfreeze it for every tournament worth 10k gold.
You also should determine, before you start raising your monster, what you want to do with them. If you plan to Battle a monster a lot you should be prepared for it to have a significantly shorter lifespan since Battle reduces a monster's life by 1 week each time you participate in a tournament. This makes it really difficult to try and max more than one of a monster's stats in a single life time.
I would recommend you have a monster you use for Battle and other monsters which you focus on raising and maxing out as many stats as possible. The reason for this is simple, when a monster dies in Monster Rancher 3 you'll be given a heart with their name on it. When you give this heart to a new Monster it will give them 20% of your old Monster's total stats as well as a new Tech. An example, if your recently deceased Monster had 999 Speed and 999 Power he'd pass on 199 Speed and 199 Power to the new Monster through his heart.
Stats, What They Do & Their Importance
This section is meant for anyone new with the Monster Rancher series or someone whose forgotten what many of the different stats do in the game and their importance. Fans of the series will notice that this Monster Rancher does away with the Skill stat, Speed replaces Skill in this regard. As said below, the two most important stats in this game are still your main damaging stat and Speed.
#1 Power: Power increases the damage done by Pow (Beat) based moves. If your monster primarily uses melee/physical moves, Power is the most important stat for them.
#1 Intelligence: Intelligence increases the damage done by Int (Water, Wind, Heart etc) based moves. If your monster uses primarily casting moves, Intelligence is the most important stat for them.
#2 Speed: Helps your monster dodge and hit their targets. This is the second most important stat for you to get for all monsters, with high power/int and high speed you can win every battle - it all comes down to luck which is basically all this game is about anyway. Battles are entirely RNG.
#3 Life: This increases your monster's total Life. Both Life and Defense will make your monster live longer and both in my opinion are equally valuable.
#3 Defense: This decreases the damage your monster takes from both Magic and Physical spells. As said with Life, both Defense and Life are equally valuable in my eyes.
Best Path for Raising an Intelligence based Monster
There's two different routes for Intelligence based monsters in the game, the one I am showing you below is best for monsters whom exclusively use Int based moves. If your monster uses some Power moves and some Int moves you may not want to drop their Power stat to 1 like I do. This poses different challenges since many trainings that raise Int which don't drop Pow drop Spd which is arguably even more important.
Best Path for Raising a Strength based Monster
Below is a quick list of the three drills that I recommend you run on your monster if Strength is their primary damage dealing stat. The first is in Takrama and I typically stay there until I get 600 Strength then I move to Goat and stay there until I get 800 or so Speed at which point I move to Kalaragi where I either do Chase until I max Speed then do Dive and raise Defense until the monster dies... Or I do the Chase drill while Battling the monster.
Notes
These are random tidbits of information that I picked up while raising dozens of different monsters.
1. Running all light drills in Morx and never letting your monster get tired allows them to live until 5 years old.
2. Only running drills that raise two stats and lower one will drive your monster into an early grave.
3. During a monster's childhood or adolescence feed it nothing but the region specific food to get it to chance species.
4. In this game feeding your monster Diet Grass is basically a requirement. They'll get fat in no time if you don't.