There is a formula that Christ outlines here -
Answer = A + B + C
"Come after me" = Deny yourself (disown/abstain) + take up your cross daily (self-dying, calling, commission) + Follow Christ (not be a "fan", not have Jesus follow you)
If anyone would come after me is the answer to the equation. The rest of the arrangement is how one gets to that point. We are encouraged to deny ourselves. In the original Greek that meant disowning one's self or abstaining from selfish desires. We all have to take care of daily needs, but when it comes to time and energies correctly spent - Christ gets first dibs, then others, finally yourself.
We are also encouraged as part of the equation to take up our cross daily. Think about the audience Jesus was speaking to - His disciples. Taking up a cross AND on a daily basis did not appeal to anyone, especially observing Roman crucifixions all over the countryside as a normal control factor for the Empire. It is strong imagery meant for a strong message. From the time Jesus arrived as a little baby onward, He was always working on His calling - to die for the sins of the world - His Cross. So we see in this verse that Jesus requests His would-be Believers that their burden or calling for God be picked up daily, even if it somehow means death! Something that is going to cost you and me because He required of us to be "living sacrifices" (Romans 12:1) - walking around in service to God and others in some form or fashion. This is the kind of payment it cost Him and He is not going to ask more than He was willing to give.
The personal qualifier for taking up your cross comes in verse 24 - "whoever loses his life for me will save it". I believe this verse can mean both who literally loses his life and who humbles his life. You don't see many new Christian converts going around killing themselves. However, one is expected to accept that risk along life's journey as we humble ourselves in the meantime.
Finally, we are encouraged to follow Christ. We have denied ourselves, taken up our cross, and now it is time to actually follow Jesus. The original Greek word for "follow" here is akoloutheō. This means properly to be in the same way with, that is, to accompany (specifically as a disciple). Let us get something straight right away - you can't follow someone if you are in the lead. We need to follow Him daily, He does not follow us around and adhere to our agenda. That is not a follower. Also, someone who talks a great talk about the faith but does not show anything - those individuals are called fans. They may have good intentions, can cheer for others, but never have any inclination of getting involved. This is not a follower. In Christianity there are many fans. Jesus did not call the Disciples to come be a fan of Himself. No. Jesus said, "Come, and follow me..." (Matthew 4:19)
Are you considering becoming a Christian or going after Christ with more passion as a Believer? Then first read all that Christ Jesus did for you in Scripture and then you can disown yourself, take up your calling, and discover what it means to follow Him. He died for me, I'll live for Him.
Take Up Your Cross by Brooklyn Tabernacle Choir (Turn up your speakers and hit play)