As John MacArthur said in a recent radio broadcast, Satan and his minions are involved 24/7/365 in doing whatever is in their power to "kill, steal, and destroy" what God is doing in the world through His church and through the servants of His church.
Paul, being as he was on the cusp of doing great things for God by reaching out to the Gentiles with the redemptive message of the Gospel, was not immune from Satanic and demonic hindrances, and if Paul said "Satan hindered us," he was neither copping out nor engaging in the "blame game," but was rather stating a fact.
Satan and his demons and their ways are devious, evil, implacable, insidious, corrupt, immoral, crafty, wily, and much too powerful for even the most spiritual of believers to go "head to head" with them. Moreover, any attempt to "dabble" in the occult is perhaps almost as dangerous as attempting to engage occult forces in combat, which Paul was wise not to do.
The best we can do as believers is to put on the complete armor of God and simply stand against Satan in the power with which God supplies us (Ephesians 6:10 ff.). Prayer, too, can often be effective, particularly when combined with fasting (see Matthew 17:21 and Mark 9:29).
Perhaps God has gifted some of His servants with the gift and responsibility to cast out demons. Back to John MacArthur for a moment, he tells of a time when he came into the presence of a person who was demonized, and the demons communicated with John, telling him they knew who he was and that they wanted him to leave. From that one experience alone, John realized he was out-classed and out-gunned by the devil's demons, and he refused to engage them in battle. The wise man that he is, he, like Michael the archangel who disputed with the devil regarding the body of Moses, said simply, "The Lord rebuke you" and left the scene (see Jude, vv.9,10). We would be wise to do the same, unless God has called us to a ministry of what the Roman Catholic church calls exorcism. In the New Testament it is called simply "casting out demons" (see the list here).