Jesus and his disciples left the synagogue and entered the home of Peter and Andrew. At the time, Peter’s mother-in-law had a high fever. Informed about this, Jesus took hold of her hand, raised her up, and “rebuked the fever.” Liberated from the fever and with her full strength restored, the mother-in-law got up and began to serve Jesus and the others, probably providing food for them. (Matthew 8:14, 15; Mark 1:29-31; Luke 4:38, 39)
In the evening, after the Sabbath had ended at sundown, people came to the home, bringing the sick and those they believed were suffering from demon possession. Jesus healed the sick and those who were possessed, many of whom screamed, “You are the Son of God.” He, however, did not permit them to speak, as they knew him to be the Christ. The cures Jesus effected fulfilled the words of Isaiah’s prophecy (53:4), “He took our infirmities [away] and carried our diseases.” (Matthew 8:16, 17; Mark 1:32-34; Luke 4:40, 41)
It appears that Jesus stayed overnight at Peter’s home and then, while it was still dark, got up early in the morning. He then headed for an isolated area, where he could pray. (Mark 1:35)
In the meantime, people probably came to the house, looking for Jesus. Peter and the other disciples then searched for him. Upon finding him, Peter said, “All are seeking you.” (Mark 1:36, 37) According to Luke’s account, quite a number of people seem to have followed the disciples and tried to prevent Jesus from leaving Capernaum. He, however, replied, “I must proclaim the glad tidings of the kingdom of God also in other cities, because for this [reason] I have been sent.” (Luke 4:42, 43) In Mark’s account, Jesus directed his words to Peter and the other disciples, “Let us go elsewhere, into the neighboring towns, that there also I may preach; for this [reason] I have come.” (Mark 1:38) Accompanied by his disciples, he went throughout Galilee, teaching in the synagogues, preaching the glad tidings of the kingdom, curing every disease and ailment among the people, and liberating many from the power of the demons. (Matthew 4:23; Mark 1:39)
The inhabitants of Capernaum observed the Sabbath and so waited until after it ended to bring the sick and those suffering in other ways.
The quotation of Isaiah 53:4 in Matthew 8:17 does not follow the wording of the extant Septuagint text, which reads, “This one carries our sins and undergoes pain for us.” The quotation in Matthew does, however, agree with the reading of the Masoretic Text and the Dead Sea Scroll of Isaiah. “Surely he has borne our sicknesses and carried our pains.” The extant Septuagint text is somewhat closer to the thought expressed in the Targum of Isaiah, “He shall pray for our transgressions and our iniquities.”
According to the reading of Luke 4:44 in the oldest manuscripts, Jesus preached in the “synagogues of Judea.” If this is the original reading, “synagogues of Judea” may designate the synagogues in the land where the Jews lived and not specifically those in the region known as Judea, situated on the southern border of Samaria. Later manuscripts read “Galilee.”
See http://bibleplaces.com/capernaum.htm for pictures of and comments about Capernaum.