Paul longed to visit the believers in Rome because he wanted them to grow up in Christ.  He knew that they had started well, they were trusting Jesus and had formed a church which was alive in Christ (Romans 1:8).  We do not know what 'spiritual gifts' he wanted to give them, but he knew they would enable them to be more confident in Jesus.  

There is nothing parents hope to see more than their children growing up, being who they are in the context of the family.  In the same way, Paul longed for all the churches to grow up into God’s family (1 Thessalonians 2:11-12).  Their growth would be his joy (Hebrews 13:17), and his joy in their fellowship would be greatly encouraging to them.  But Paul wanted to remove any doubt about his personal care for them; his absence was not due to lack of desire.  Perhaps as in the delayed visit to Thessalonica, Satan had prevented him from visiting Rome (1 Thessalonians 2:18).  But it was allowed by the Lord - so that we would have the letters to Thessalonica and Rome passed on to us, and many generations before us, to build up our faith in Jesus also.

Paul also wanted to see a 'harvest' in and through the church in Rome.  The seed of God's Word grows inside people to produce fruit (Luke 8:5-15): new spiritual life as unbelievers are born again  and Christ-like qualities of character in believers.  Paul's ambition was to see the life of Christ grow and spread everywhere.  That is why he wanted to go to Rome, the heart of an Empire of about 50 million people, to preach the gospel.  He fully expected that whenever he spoke of Christ, people would respond and be changed by God's Spirit (1 Corinthians 9:19-22).

Spiritual ambition should be an important part of a Christian's life - a strong desire for God to be at work in people's lives, and a willingness to work with Him to enable His kingdom to grow.  Alas many of our desires are focussed on our own interests, hoping that our own little 'kingdoms' will be built up - they are selfish ambitions.  So much giftedness, money and energy go into making ourselves important, when it could be used to help produce a spiritual harvest. Everyone who claims to follow Jesus should regularly assess their ambitions. Are yours selfish or spiritual? The test will be the sort of harvest which comes from your life.