I wouldn't be too concerned about getting new monsters until your current one is getting old and is ready to put on deep freeze for breeding. You'll just get yourself a lot of frustration and such otherwise.
Essentially combining things via the freezer chamber is breeding with the game getting around having to talk about the birds and the bees. The monster you get will be a baby rather than an elderly combination of the parents.
also, keep in mind the traits of various monsters as you poke around with them, because you might want to have a certain combination of monsters for something you want to try.
I'm sure when you got your monster you saw in the species something like Birdon Goblin or some other combination of names of species. This is the major/minor species combination. Most of a monster's traits, such as shape and moves, are determined by the dominant monster trait, while a few things, like personality, a few passive skills, and sometimes an attack or two come from the minor species.
a couple examples from monster rancher to illustrate a couple things.
pixies were a base type in the monster rancher series and were all pretty much attractive adult female creatures. They were known for high intelligence, skill (accuracy) and speed and their attacks were almost entirely magical.
If you combined a pixie with a Worm (exactly how it sounds) you would get a mothlike winged woman called a "nightflyer." In appearance this creature would have the basic shape of a pixie, though with some insectile traits, also having some coloration similar to the worm.
Worms were known for being pretty fast, and relatively accurate as well, but they were tougher than pixies and considerably more stupid. Most Nightflyers still had high intelligence, but not as high as a basic pixie, and had crazy-high speed and skill, and their skills, because they combined the pixie and worm were more balanced, being propped up a little by what the worm was good at, while also sacrificing the intelligence stat of the pixie a little. The nightflyers also got a couple unique attacks late in their move sets that were unique to them or at least were either very hard for normal pixies to get or because of the nightflyers' different stat bonuses, far more useful to the nightflyer.
another example of how species can interact.
Golems were huge stone men who hit really hard but were slow and stupid. There were box-like creatures called koropandoras that were trios of boxes that worked todether as a single monster. If you combined the two, sometimes you would get a creature that was shaped like a golem (being mostly golem) and fighting as one, but actualy being made of a small army of koropandoras haphazardly stuck together.
Koropandoras had a special power that would sometimes come up in battle called "Unity" (I never figured out what this meant but I know from experience it was good) which made sense because the monster was 3 of them working as one. Because the weird golem offshoot had koropandora in it, even though it technically wasn't one, it would also sometimes get "unity" in battle as well.
It still had the golem attacks, for the most part had the golem stats, but it had a personality more like koropandoras and could access a couple of their passive skills and random buffs at times.