The fossil record suggests that new species arise quickly, on geological scales. The evidence is consistent with even one generation. Even classic examples of gradual speciation (rare as they are) like the horse, when looked at closely show that there are spurts, where for example a new toe arises. The current explanation for the so called punctuated equilibrium model of speciation is that even though new functions and morphologies require changes in hundreds of different genes in a coordinated fashion, these changes do not occur by mutations in those genes (that would have 0 probability to happen over a small timescale), but by mutations in a few so called master genes which regulate 100s of different genes, or drastic chromosomal events like duplicating chromosome number (hybridization between diploid plant species to form tetraploid progeny, something that does not seem to happen in animals), large deletions, partial duplications, unbalanced translocations, etc. There might be other explanations like morphic fields, but there is not enough evidence for these to be accepted in mainstream evolutionary biology.
The other necessary ingredient for speciation to occur quickly is reproductive isolation. This means that either a small group is unable to mate with the larger group from which it originated, or that the larger group is reduced to a small group by environmental forces like famines, climate change, invasive species, etc. The second possibility is not usually considered an example of reproductive isolation, but since it is due to the same mechanism (which I will explain next), I consider it to be a special case of reproductive isolation. If a species does not split into two (and the process can iterate) but becomes a new species without a split(I think this is called chrono-speciation, which might have happened with some of our hominid ancestors), it requires a small population, for reasons which will become apparent below.
The reason why reproductive isolation is necessary is because
1. selection for beneficial mutations is not the only force in evolution and also
2. a mutation may not be beneficial until the right environment is reached.
Here is an explanation of each of these:
1. Another force, known as genetic drift occurs because many random events participate in the fixing of a mutation in a population, including
A. achieving homozygosity (most mutations are recessive--meaning their phenotype is invisible in heterozygotes) and so either at least two people, a male and a female in sexually reproducing populations, have to get the mutation simultaneously in their germlines, or some incest for which taboos have evolved in primate populations, has to occur) which is dependent on getting the heterozygotes to find each other, mate, and produce viable progeny.
B. Random death of the carriers of the mutation before sufficient reproduction,
C. Other possibilities, related to reason 2 below, where a mutation might diffuse through a population with no benefit to its carriers until the right time (which could be the occurence of another complementary mutation, or a change in the environment). This could be thought of as tunneling through the fitness mountain (or going through the Mines of Moria in LOTR), where the phenotype is invisible until you get to the other side, where another valley exists. Most of the tunneling process could be gradual, but the breaking through to the other valley is quick.
2. The right environment could be either external, or internal, as in all the other genes that cooperate and compete with the mutated gene. In fact, on short timescales, a mutation which would be beneficial in the long run might be actually harmful in the short run, or at least neutral. These two possibilities can be illustrated by the following model. Imagine that a village of blind people, nestled in a valley surrounded by mountains has an earthquake, which shifts the landscape. The inhabitants are most content at the lowest possible elevation, but now they are no longer at that lowest elevation. Some of them might be at a local minimum, having moved away from the main population when the whole valley was flat (a mutation which was neutral previously, and now is advantageous due to a shift in the environment). Most of those are not in the valley (of comparable or lower elevation to what they had previously) on the other side of the mountains yet because that is still too far away--neutral mutations give rise to different strains, or may be totally invisible, but almost by definition (or what should be the definition) of a species, drastic or at least very specific changes are necessary to get to the other side of the mountain. Those few who try drastic (or very specific, lucky but not drastic) shifts, (either in their internal memetic space or in their external geographical space) are selected against because they have to undergo some extra hardship to get to the mountain pass--the mutations are disadvantageous in the short run. Or they fail not only because of extra hardship at the higher elevations, but because their efforts are constantly diluted by the majority of the villagers who would rather be more comfortable (more comfortable than if they go up, but less comfortable than if they stay put or even go down to a valley that is still much higher than what they had before) and have evolved a conservatism that has served them well in times before the earthquake. A few may either have been lucky enough to be on the other side of the new mountain pass before the earthquake so that they are memetically isolated from the rest of the villagers, or they can persevere and make it to the mountain pass despite hardship. I suppose the neutral invisible memetic mutations that do not cause extra hardship could keep happening to people in the main village even after the earthquake, but there is not much time. As the population dwindles in the main village, the force of drift dwindles as well.
I part with mainstream evolutionary theory in that I consider reproductive isolation to be a necessary but not sufficient condition for speciation. Perhaps the sufficient condition is a mutation in a master gene (followed by rolling downhill in negative fitness space towards a new valley), which also implies genetic isolation, as the two populations are now separated by a mountainpass. Anytime reproductive isolation has been achieved in the lab at a genetic level (so not just by separating the two populations), the result is closely related strains. Perhaps over eons they will develop significant morphological and functional differences, but since this doesn't seem to happen in nature, I doubt it would happen in a lab, even with the acceleration of mutation rates offered by radiation or chemical mutagens. My bet is that drastic genetic events will occur quickly (due to a mutation in a master gene), followed by further slow divergence (drift and divergence in the new valley).
As far as how these things concern us, people participating in cultural evolution, unlike organisms participating in biological evolution, we are not totally blind, but neither can we see very far into the realm of possibilities ahead. I agree with JMG that the more experiments/expeditions up the mountain or through the mountain are tried the better chance of finding mountain passes. Small, gradual changes are not likely to get us to another valley before being swamped out by the mainstream culture, unless we get lucky and find just the right master memes. Memetic isolation, despite seeming harsh, might be advantageous (do we have to give up the internet, movies, literature, economic interaction)--how much isolation is necessary? there has to be some interaction with the rest of the ecosystem and the analogy of the mainstream culture being the parent population breaks down--it is also an analogous major part of the ecosystem). Otherwise, we can wait for a large population decrease in order to make some progress towards a mountain pass, a possibility I do not savor given the alternative.