This is a very interesting question! Caesar lived from the period of 100 to 44 B.C.E, but for the purposes of answering your question I'm going to focus on the period of 100 B.C.E to 15 C.E. This period covers the late Roman Republic, including the dictatorship of Sulla, through the death of Augustus Caesar.
There are actually a wide variety of specimens discovered by archaeologists; some of them are remarkably well-preserved and can be seen in museums like the Metropolitan Museum of Art, the Royal Ontario Museum of Archaeology, the Museum of Naples, and the Museum of Cairo, for example.
Some common modern cooking utensils, such as the frying pan and saucepan, date back to well before the Roman Empire period, and there are examples of Greek versions dating to several centuries before Caesar's era. These utensils were primarily terracotta; B.A. Sparkes notes the reason for this being "clay was cheap, bronze dear".^1
The Greeks also used terracotta vessels for storage of grains, cereals, and other dry foods, and jars which could be sealed for wine - these ancient wine jars even used corks! Examples of bronze cookware do exist, but were the province of the wealthy. A banquet, for example, would probably have included bronze serving dishes.
The Romans used similar cookware to the Greeks, and Roman cuisine was much influenced by its Greek cousin. Grains, vegetables, and cereals were the chief staples of the Greek diet, and fish were consumed much moreso than red meat; contemporary cookware reflects this reality^(2).
Especially for the common people of Rome, bread was extremely important to their diet. Riots took place when the price of bread rose too high; this is where Juvenal's famous "bread and circuses" quote originates from. Just like today, there were differences in diet for the poor and the wealthy.
By the time Julius Caesar was an adult, a wide terracotta bowl called the olla was one of the chief examples of Roman cookware, used in the preparation of porridge, vegetables, and meats^3 by Roman cooks. There are also examples of bronze kettles and strainers, long-handled ladles, and frying pans made of bronze or iron (including, remarkably, frying pans with moving and folding handles^(4)).
Knives of all sorts were also in use; cleavers and butcher's knives for cutting meat, smaller knives for chopping vegetables and the like, etc. Forks did exist, but were much larger and primarily used for preparing meat. Meat itself was typically served pre-cut.
There were table spoons used for soups, etc., but other table utensils did not spread until later in the Roman Empire; table forks became common around the 5^th century C.E. in the Eastern Roman (Byzantine) Empire.
However, smaller utensils such as table forks and knives were not used, and the Romans of this era "ate with their hands"^(5). Evidence for this can be found in Ovid's Ars Amatoria, a poetic guide to dating and courtship, which includes a passage on eating "daintily" with one's hands and not "smearing grease on one's face"^(6).
The lack of table utensils is at least in part due to the existence of the triclinium, the Roman formal dining room where Romans would lie on their sides on couches similar to chaise lounges, being served food by slaves; this eating position meant wealthy Romans ate with one hand, precluding the use of a knife and fork. Thanks to remarkable finds at Pompeii and Herculaneum, we have very-well preserved evidence of the triclinium.
So to sum it up, during the late Republic and early Empire, cookware had many similarity to modern cooking. There were large, broad pots and bowls, frying pans, saucepans, and many of the same kitchen utensils that we see today, such as kettles, ladles, colanders, and strainers. However, they did not have table forks, and only occasionally used table knives.
If you're interested in learning more about Roman cuisine, Celsus's On Medicine has many descriptions of Roman cooking, primarily to describe various foods and whether they should be provided to individuals with certain illnesses. There is also a remarkable cookbook by Apicius dating to the late fourth or early fifth century C.E., which contains recipes for roast chicken, oysters, peas in the mod, and even more exotic foods like roast ostrich and stuffed dormouse; this cookbook, however, dates to approximately five hundred years after the death of Caesar.
Edit: Thanks for the gold, kind stranger!
Notes:
Sparkes, B. A. "The Greek Kitchen." The Journal of Hellenic Studies 82 (1962): 121-37. doi:10.2307/628548. Page 124.
Sparkes, B.A. "The Greek Kitchen." Pages 122-123.
Harcum, Cornelia G. "Roman Cooking Utensils in the Royal Ontario Museum of Archaeology." American Journal of Archaeology 25, no. 1 (1921): 37-54. doi:10.2307/497888. Pages 37-38.
Harcum, Cornelia G. "Roman Cooking Utensils." Page 44.
J. H. B. "Knives and Forks." The Metropolitan Museum of Art Bulletin 2, no. 9 (1907): 154-56. doi:10.2307/3253056. Page 155.
Ovid. Ars Amatoria. Book III, Chap. XVII. Trans. A.S. Kline.
Sources:
Apicius. Cooking and Dining in Imperial Rome. Trans. Joseph Dommers Vehling, 2009. http://www.gutenberg.org/files/29728/29728-h/29728-h.htm#bkviii_chix
Celsus, Aulus Cornelius. A Translation of the Eight Books of Aulus Cornelius Celsus on Medicine. Trans. Simpkin and Marshal, 1831.
Harcum, Cornelia G. “Roman Cooking Utensils in the Royal Ontario Museum of Archaeology.” American Journal of Archaeology, vol. 25, no. 1, 1921, pp. 37–54., www.jstor.org/stable/497888.
J. H. B. “Knives and Forks.” The Metropolitan Museum of Art Bulletin, vol. 2, no. 9, 1907, pp. 154–156., www.jstor.org/stable/3253056.
Gisela M. A. Richter. “Greek and Roman Bronzes: Recent Acquisitions.” The Metropolitan Museum of Art Bulletin, vol. 19, no. 3, 1924, pp. 68–72., www.jstor.org/stable/3254937.
Sparkes, B. A. “The Greek Kitchen.” The Journal of Hellenic Studies, vol. 82, 1962, pp. 121–137., www.jstor.org/stable/628548.