"When they brought the colt to Jesus and threw their cloaks over it, he sat on it. Many people spread their cloaks on the road, while others spread branches they had cut in the fields." NIV translation
As Jesus rode the donkey people put their coats down in front and others put down branches. This was a sign of submission to a king as most people did not ride donkeys. In times of peace, a king would ride a donkey while in war he would ride a horse.
"Those who went ahead and those who followed shouted, 'Hosanna!' 'Blessed is he who comes in the name of the Lord!' 'Blessed is the coming kingdom of our father David!' 'Hosanna in the highest heaven!'" NIV translation
Hosanna originally meant something like "save we pray" but it became a term of praise. The people were shouting this and also a quote from Psalm 118. The people acknowledge that there is praise on earth and in heaven as Jesus rides in as King.
"Jesus entered Jerusalem and went into the temple courts. He looked at everything, but since it was already late, he went out to Bethany with the Twelve." NIV translation
We see that this first trip into Jerusalem was simply to make a declaration that the King had come. It was late when they arrived so they turned around and went back out to Bethany.
"The next day as they were leaving Bethany, Jesus was hungry. Seeing in the distance a fig tree in leaf, he went to find out if it had any fruit. When he reached it, he found nothing but leaves, because it was not the season for figs. Then he said to the tree, 'May no one ever eat fruit from you again.' And his disciples heard him say it." NIV translation
As they were heading back to Jerusalem, Jesus saw a fig tree that was full of leaves. This would have been unusual because, at this time of year, they just begin putting on leaves. As Jesus went to taker a closer look, He saw that this tree that was full of leaves did not have any fruit on it. He would not have been expecting to see ripe figs on the tree as it was not the proper season but there were not even the beginning of fruit on the tree. Jesus "cursed" the tree by proclaiming that nobody would ever eat fruit from it again.