The first part of verse one really says it all about spiritual gifts : "Follow the way of love and eagerly desire the spiritual gifts". They are definitely a fundamental part of the the Christian life, used in the right context. "Love" here essentially means "for the building up of others". Paul then talks about the difference between tongues and prophecy, and while tongues are good, prophecy is even better because it builds up the church.
When we speak in tongues, unless someone has the gift of interpretation, no one (including the one speaking in tongues) has a clue what the person is speaking about.
Tongues is primarily for the building up of the individual, it does not do a great deal of good for others. When we pray in tongues we pray with our spirit, not with our mind. What does this mean? Well consider this. With normal English words I can silently run through them in my mind. With tongues I cannot do this, I have to actually mouth the words. I can do it quietly, but I do actually have to say them. This is because tongues are a product of the spirit not the mind.
Now some may say what is the point of this meaningless gibberish. The point is that it edifies my spirit. We need to build up our spirits and our minds. Paul says we should pray with our spirits and with our minds, so we should do both. At the same time we need to be aware that no one else will benefit particularly from hearings us speak in tongues.
With prophecy, on the other hand, everyone who hears benefits. It is designed for the benefit of others, for their strengthening, encouragement, and comfort.
At the end of the section Paul talks about their use in public meetings and the sort of sign they are. What does he mean? Well sometimes we seem to treat tongues as if it is some sort of magic, it isn't. In particular, praying on tongues in a public meeting is at best harmless, and worst harmful. It does not show that God is especially present! As Paul says, if an unbeliever wanders in they will just think everyone is mad (and may well be right!). The only sign it gives is convincing the unbeliever that he is shut out from the things of God and can never make any sense of them. Tongues is not a sign of God's blessing on a meeting.
Prophecy, on the other hand, is extremely useful in public meetings, and is a sign of God's presence. This time the unbeliever will make sense of it, may well be convicted because of it.
I will close with some final remarks about the questions "should everyone speak in tongues"? I believe God can give the gift to anyone, but at the same time speaking in tongues does not make you a superior Christian, and not speaking in tongues does not make you inferior. Sometimes people can get very uptight about speaking in tongues or not speaking in tongues. I would simply say "relax".