First and Second Timothy are personal letters from an older mentor (Paul) to a young pastor (Timothy). Just before our text above, Paul has called on Timothy to "wage the good warfare" (1 Tim. 1:18). How is Timothy to do this? Paul says very clearly the most important thing Timothy can do is pray. Look at Paul's language: "first of all." Out of all of Timothy's duties, prayer is the priority. Paul puts prayer even before preaching.
Paul does not desire just a Lord's Prayer tossed up to heaven before bedtime--Paul urges Timothy to thank God and pray "for all people." This is an intense duty. In fact, this is "warfare." This is the most important thing you as a Christian can do to further God's kingdom. If Paul's (God-breathed) urging is not enough to make you pray, look at the result of prayer. If I went to all Christians in the world and offered them "a peaceful and quiet life, godly and dignified in every way" in exchange for money; every one would jump at the chance. But God does not desire money, he makes this the result of prayer. Let us not ignore the opportunity God is giving us in prayer. As James puts it so bluntly, "You do not have, because you do not ask" (James 4:2).
This, however, is not easy to apply. How can we pray and thank God for all people? That would take all our time, right? Exactly. I've often struggled with the command to "pray without ceasing" because I simply don't have that many things to pray about. But there are always people in need of prayer around us. Here's my plan for applying this text to my life: pray for the people who I am with. At work each day, I interact with hundreds of people--most of whom are most likely not saved. They need God's salvation or they will suffer God's judgment. They need conversion more than they need air. And, in Jesus, I have God's ear (so to speak). I can pray for each one of my coworkers and customers as I interact with them and after I see them. Outside of work, I see many other people, all for whom I should thank God and intercede. This is just my way to apply this; take it or leave it, but by all means, follow Paul's urging and and "first of all" pray.