The art of the first and second centuries AD pretty much continued the traditions of portraiture and Greek imitations. Roman artists added more use of art as propaganda to show what the emperors wanted people to know or to think. Some examples of this are the Arch of Titus and Trajan's Column.
First Style (imitating marble panels)
There was also a lot of wall painting to decorate the walls of houses during this time. The wall painting of the first century AD is sometimes divided into four different styles, mainly because of the many different styles of wall painting that were found at Pompeii. In the first style, the fresco painting on the walls of houses is meant to look like marble panels (but it's a lot cheaper than marble panels!).
Second style (with garlands added)
In the Second Style, the artists begin to add little things to the imitation marble panels in their paintings. This one has a garland. Other paintings have fruit, or flowers, or birds perched here and there.
Third Style (with people and scenes)
Third Style Roman wall painting takes this idea further by adding whole scenes to the walls. Here in this painting from the Villa of the Mysteries at Pompeii you can see full-size people talking to each other and sitting on chairs, as if there were another room there instead of a wall.
A Gaulish pot
Of course there are also local variations all over the Roman Empire. An empire which covered most of Europe and all around the Mediterranean could hardly have only one art style all over it. The Gauls continued their art styles from before the Romans came, and found ways to mix their old art styles with new Roman ideas. So did the Britons, and the Spanish, and the Carthaginians, and the Phoenicians, and so forth.
Rock-cut temples at Petra in Jordan
Learn by doing: draw pictures in First Style, Second Style, etc.
Later Roman art
Ancient Roman Art, by Susie Hodge (1998). Easy reading.
Roman Art: Romulus to Constantine, by Nancy and Andrew Ramage (4th Edition 2004). The standard textbook.
A Coloring Book of Ancient Rome, from Bellerophon Books (1988). Easy reading.