When I took a course on genetics and evolution, I learned that recombination and sexual reproduction is advantageous compared to asexual reproduction. Sexual reproduction allows more combinations of DNA because of recombination, allowing more variations of species, causing natural selection to "choose" different variations of species to reproduce and share their genes to help the species from dying out. On the other hand, asexual reproduction doesn't allow recombination. So species can only rely on mutational changes to keep a species safe from extinction. Therefore, why don't we see more sexually reproducing species than asexually reproducing species? Sexually reproducing species have an advantage, don't they?
When you say, why don't we see more sexually reproducing species than asexual, I presume you are referring to bacteria, protists, archaea, some fungi, etc.
Most multicellular organisims do reproduce sexually, which makes sense since an organism would need to be multicellular to have cells specialized for gametes. As you stated, the advantage is that sexual reproduction offers an advantage by allowing more rapid generation of genetic diversity, allowing adaptation to changing environments.
However, look at the single-celled organisms that reproduce asexually. Asexual reproduction is beneficial in many circumstances in that it can create individuals rapidly and in large quantities. For example, yeast can duplicate every hour under the proper conditions. For yeast, that means survival by sheer numbers. For us, it means we can make bread in a few hours.
If you believe asexually reproducing organisms are at a disadvantage due to lack of genetic diversity, consider bacteria and antibiotic resistance. The rapid generation times allow mutations to be selected for in particular circumstances. Penicillin was introduced in 1941. Two years after this landmark event, strains of Penicillin-resistant S. aureus had already emerged. Now, perhaps 2% of Staph. aureus (SA) is susceptible to penicillin, and some SA are "superbugs" with multidrug resistance: good for Staph, bad for us.
Also, it is possible for single-celled organisms to "share" DNA through mechanisms sometimes likened to sexual reproduction; lateral gene transfer occurs through conjugation, transformation and transduction.
Thus, both asexually and sexually reproducing organisms are well-suited to their ecological niches.
This just in addition to @anongoodnurse's excellent answer. It was mentioned in the OP that asexual organisms do not undergo recombination; this is not true. Recombination is used for integration of foreign DNA into prokaryotic chromosomes and for repair.
Also, don't underestimate the power of mutational change in rapidly reproducing organisms. While mutation rate is low, consider the effect when it is applied to millions of individuals.
Finally, just an interesting example of the ability of bacteria to evolve. Many organisms can rearrange foreign DNA for their benefit. I will mention the specific example (because it's what I work with) of transforming B subtilis with a plasmid containing an antibiotic resistance marker and somewhat toxic gene. Often you will find transformed bacteria that have rearranged the plasmid to delete or inactivate the toxic gene while still retaining the resistance gene. Thus they can still grow in the selective medium and manage to be of no use to me whatsoever!
Sexual reproduction probably evolved because an asexually reproducing species of bacteria evolved an evolutionary stable strategy and then there was a rapid change in the environment and only the mutant individuals that could reproduce sexually could evolve fast enough to survive which later evolved into sexually reproducing animals. Also, sexually reproducing species are more likely to be able to evolve fast enough to avoid extinction when there's a rapidly changing environment. However, the true evolutionary stable strategy for any species probably has the species reproduce only by parthenogenesis. Indeed, some bacteria probably didn’t evolve to reproduce sexually and then evolved into an asexually reproducing animal species which still exists like the New Mexico whiptail and Mycocepurus smithii.
If an asexually reproducing species happens to have the right genome, it will already be adapted to all those future rapidly changing environments that will happen like the spread of a deadly parasite assuming the Sun won't get brighter or die. The ones with a slightly faster mutation rate will very slowly outcompete the ones with a slower mutation rate because they can evolve the trait that's best for the current environment faster but once the next sudden change in environment occurs, the ones with a lower mutation rate will survive because they remained adapted to the change. After enough sudden changes to the environment occurr, the ones that survived the most recent sudden change in the environment will be the ones that had a genome that prediced that that environment could occur already was adapted to it because it was adapted to the sudden changes that already happened and there is a pattern in which changes in the environment already occurred.
Another advantage in asexual reproduction is that a mutant asexually reproducing individual who happens to have the right genome in a sexually reproducing species could rear 2 kids and pass all their genes onto them if the species was already at an evolutionary stable size. However, once the asexually reproducing individuals outcompete the sexually reproducing ones and reach a new evolutionary stable size, each individual on average could only rear 1 kid.
Another advantage in asexual reproduction is that any mutation that's advantageous for an offspring is also advantageous for their off spring and the mutation rate is very low making the strength of genetic drift very weak so there would probably be less of a tendency that the more of one advantageous trait there's genetic drift against that a species has, the less of another advantageous trait there's genetic drift against it can evolve to have. For example, an asexually reproducing human could outcompete all humans other than her descendants and then evolve a genome that gives them immunity to malaria without the possibility of sickle cell anemia. An sexually reproducing species if it evolved resistance to enough past diseases doesn't necessarily have to be more prone to the spread of a deadly parasite than a sexually reproducing species. Frequency dependent selection could easily select for mutations that make some individuals different from others in order to have resistance to a deadly parasite that already evolved to spread through some individuals.
The smaller an animal is, the faster it can run without ever getting too hot because the faster it can get rid of heat. However if it exceeds the speed it can go without ever getting too hot, then the faster it runs, the shorter the distance it can run before it collapses from heat exhaustion. A larger animal that exceeds the speed it can run without ever getting too hot can run longer at that speed before it gets too hot than a smaller animal running at the same speed.
I read that a wolf never gets too hot in its chase if it's not too hot outside so a mutant asexually reproducing wolf probably would have a huge evolutionary advantage for chasing rabbits as long as the air's not above 30°C. It would be able to evolve into a poikilotherm so that it can do away with having fur so that it can get rid of heat faster so that it can chase rabbits faster so that they'll be less likely to escape and to burn less energy when it's cold out.
A cheetah on the other hand builds up heat way faster than it can get rid of it during a chase so a mutant asexually reproducing cheetah would probably also have a hug advantage. It could evolve to be a poikilotherm so that it can have a lower core temperature at the beginning of a chase during times the air is colder so that it can chase its prey for a longer time before it gets too hot to continue the chase, and also evolve to not have fur so that it can get rid of heat faster after a chase so that if a prey animal happens to come sooner after its chase, it can chase it. That's because the expected amount of energy it will gain from continuing the chase is larger than the expected energy it will burn in continuing the chase. I think they could actually get rid of heat way faster because they could walk to speed up the loss of heat after a chase because the air would be freely circulating around them.