What are some similarities between Jesus and Joshua?
The above answer gives some great point. Here are a few others that will help you.
First, it is good to keep in mind that the New Testament writers are conscious about the fact that there are previous scriptures. So, they take themes and persons from the Hebrew Bible and connect them to Jesus. So, we can say that there is continuity between the Hebrew Bible and the New Testament. From the perspective of the New Testament, Jesus is the fulfillment of Joshua. Or we can say that Joshua looks forward to the ultimate Joshua, who is Jesus.
For example, Jesus is the great leader who leads his people into the true "Promised Land." Jesus is the one who vanquishes his enemies at the second coming, according to the New Testament. Jesus is the one who gives his people true rest (Hebrews 4). Jesus is the one who truly saves his people from the final enemy - sin and death.
Hebrews 4, in the New Testament, is by far the most important passage when it comes to the connection between Jesus and Joshua. I will add a few links that will help you naviagate this chapter.
The first here to mention is their names.
Joshua also rendered in Hebrew as 'Yehoshua' meaning "Yahweh is Salvation" or Jehovah is Salvation.
The name Jesus is a Greek name also known as 'Yehoshua' meaning the same thing as "Yahweh is Salvation".
The Greeks even call Joshua Jesus. But to differentiate them they call Joshua 'Jesus the son of Nun' and they call Jesus, Jesus Christ.
Joshua is the one that led the Israelites into Canaan 'The Promised Land' and Jesus will leads us into the "promised land" of the Kingdom of God or the Kingdom of the Heavens.
Moses Law called the Mosaic Law could not lead the Israelites into the Promised Land but Joshua did. So also, Jesus the mediator of the Law Covenant will lead Christians into Heaven.
Joshua was appointed/anointed by God to lead His people into God's rest, so also according to Hebrews 4:8-10 identifies Jesus as a better Joshua, as Joshua led Israel into the rest of Canaan, but Jesus leads the people of God "into a place of rest", salvation.
Joshua was bold in delivering the message of truth about what he saw (as a spy), so also Jesus was bold in delivering his message about the good things of the kingdom even in face of the religious leaders of that time.
Joshua when about to die gathered the people and told them to follow the way of God; Jesus when about to die gathered his disciples and laid down the way to follow "God's Way.