When a germ becomes penicillin-resistant, the germ has evolved. That event may not be very exciting, but it does meet the definition of evolution. Any biologist can make a bacteria evolve in about a day or so. It's easy.
A much-used example is the speckled moth. In 1848, 98 percent of these moths were gray, and the rest were black. Then the Industrial Revolution put a lot of soot on Britain's trees. Being light colored was now dangerous to a moth that lived on tree trunks. Birds could see them too easily.
By 1898, only 5 percent of the moths were gray. More recently, the air pollution laws have cleaned things up, and gray is once again predominant. This meets the definition of evolution. The frequency of an allele changed (and then changed again).
So, evolution does happen. Notice that that conclusion doesn't depend on any theory. There's a definition, and there are observations that meet the definition. That means that the process of evolution is a fact. Theory only comes in when we try to explain the situation. Darwinians pointed out that a hard-to-see moth has a better chance of having descendants. That was the Darwinian explanation: that was his theory, applied to this case.
(Creationist Jonathan Wells has published an attack on this example. It should be noted that he only disputed the experiments that showed Darwinian explanations to be the correct explanation. No one disputes that the moths did change.)
Some Creationists make a distinction between micro-evolution and macro-evolution. They concede that small changes occur, but they argue that beneficial mutations do not happen, or that small changes do not add up over time. They say that there are limits or barriers to change. So, each kind would be a separate creation, and micro-evolution applies within a kind.
Unfortunately, kind is not well defined. Different Creationist authors have argued that specific creatures were or weren't the same kind. (These discussions are often tied to arguments about the occupants of the Ark.) And there doesn't seem to be any real system. Dr. Gish took chimpanzees and gorillas as separate kinds, but he took 850 species of bat to be one kind instead of 850 kinds. Without good definitions, it isn't clear if these barriers to change are supposed to prevent speciation, or allow it.
In any case, the process of evolution is an observed fact, and the theory of evolution has been successful in explaining that. The Creation/Evolution dispute is actually about common descent - the question of whether today's creatures have the same ancestor(s).