What is the kingdom of God and what does it mean to "further the kingdom"? Those are very big questions that would take at least a sermon or two to explain but I want to give you a brief answer here for you to think about and for you to study on your own.
The Kingdom of God is not heaven, it is God's system or God's government; in other words, it is the way God operates and the way He wants you to operate. To further the kingdom simply means to expand God's government, God's family.
If you are a child of God that means you are part of His government or family and He wants you to help expand it by EVERYTHING you do. It does mean to go and preach the gospel, it does mean to do missionary work, it does mean to visit the prisoners, feed the poor, lay hands on the sick, cast out demons, disciple people, give financially, love your brother and a whole list of other activities.
It also means when you are at work to show God's love, walk in His ways, and honor your boss. It means being a living sacrifice to Him. It means that whatever you were created to do, do it for the expansion of the kingdom. If you, like Paul are a tent maker, be the best most Godly tent maker that ever lived. Be a citizen and act like a citizen of God's kingdom and your life, by example will further the kingdom.
There is nothing more important than that. That is what God wants. God wants ALL people to come to the knowledge of Him. It is not about you, it is about others. God is all about others, getting them saved, getting them disciple, getting them healed.
Jesus last words to the disciples weren't about them or about them and their relationship with Him. That is important; don't get me wrong, you can't do these things without a solid relationship with Him. But His last words to the disciples were about other people who didn't know God. His last words were go into all the world preach the gospel, make disciples, and do the works. It's about other people and their salvation.
The gospel and the kingdom is not a me, me, me gospel. Our example, our lives, our work, our relationships are all about others. Expand the kingdom with your life and
Matt 28: 18-20, Mark 16: 14-20, Matt 4:23, Matt 6:33, Matt 13.