Mill argued for a distinction between ‘higher’ and lower pleasures. His distinction is difficult to pin down, but it more or less tracks the distinction between capacities thought to be unique to humans and those we share with other animals. Higher pleasures depend on distinctively human capacities, which have a more complex cognitive element, requiring abilities such as rational thought, self-awareness or language use. Lower pleasures, in contrast, require mere sentience. Humans and other animals alike enjoy basking in the sun, eating something tasty or having sex. Only humans engage in art, philosophy and so on.
This is not, however, correct; 'higher' and 'lower' are comparative, not categories of pleasures, as if there were a particular category of pleasure that was 'higher pleasure' and another that was lower. Pleasures are higher and lower, and some pleasures will be higher than a lot of other pleasures, but that's it. Mill is quite clear about this:
If I am asked, what I mean by difference of quality in pleasures, or what makes one pleasure more valuable than another, merely as a pleasure, except its being greater in amount, there is but one possible answer. Of two pleasures, if there be one to which all or almost all who have experience of both give a decided preference, irrespective of any feeling of moral obligation to prefer it, that is the more desirable pleasure.
Note that higher and lower are determined entirely by comparing two pleasures according to their preferability, not by classifying them into one of two groups.
Where Baggini is not far off is that human beings, in particular, prefer in this way pleasures that fulfill them more as human beings. But, being animals, we are also familiar with animal pleasures, so when we prefer more wholly human pleasures -- like those of education, art, or philosophy -- we are not making the preference out of ignorance, but out of familiarity with what we are judging. Baggini is also right that it makes more sense to treat pleasures as on a continuum -- but this is Mill's own view, and is the reason for the higher/lower terminology in the first place.
Baggini seems to think that Mill is committed to saying that animal pleasures are simply low-quality pleasures. But in fact, Mill's test is experience-based -- it doesn't prejudge what counts as lower or higher, although Mill does think it is very obvious from human experience that philosophy, poetry, and the like will often provide pleasures more consistently preferable than pleasures of food and sex.