Layers in Winter - and in Summer....
If you are visiting Abilene in the "winter" dress in layers. Many mornings I leave home and need a jacket and in the afternoon it is time for shorts. Our winters are mild. Occassionally, though, we do have ice and snow (about 1 or 2 times a year) for a couple of days. This may come in January or in even April. We do have a strong wind, so when it is cold, it feels bitterly cold.
In the summer it is hot. Mid-summer we are hitting the high 90's and low 100's. The good news is it is typically a DRY heat - which is not as oppressive as the humid heat (as in Houston). Still - carry a bottle of cold water and some sunscreen if your'e going to be out in the heat.
I personally would schedule my visit to enjoy the airconditioning of indoors (a museum or bookstore) until late in the evening (Abilene does have several nice parks).
We have very pleasant days in the spring and fall (though when these seasons actually come to Abilene may vary year to year). Mosquito Repellant
Moisturizer Throw in a lawn chair so you can spontaneously spend some time reading in one of Abilene's city parks.