Before looking at the feasts, I would just like to point out that the Sabbath is not part of this article. The Sabbath stands on its own, being part of the Ten Commandments, etc.
Both the Sabbath (and the New Moon) will be kept in new heaven and new earth.
For as the new heavens and the new earth, which I will make, shall remain before me, saith the LORD, so shall your seed and your name remain. And it shall come to pass, that from one new moon to another, and from one sabbath to another, shall all flesh come to worship before me, saith the LORD.
- Isaiah 66:22-23 (KJV)
The feasts (of the Lord) are given as part of Moses' law.
- The Passover
- The Feast of Unleavened Bread
- The Feast of Firstfruits
- The Feast of Weeks
- The Feast of Trumpets
- The Day of Atonement
- The Feast of Tabernacles
For a more detailed overview of all the feasts please read Leviticus 23.
Although not part of the original feasts, there is also Hanukkah which Jesus would have kept also.
The important thing to understand is that Moses' law was nullified at the cross when the new covenant was established.
In other words, we no longer need to keep them.
So there is no obligation to keep the feasts, but at the same time there is nothing wrong to keep them as memorial days.
For example. The passover originally came from Egypt where the angels would "pass over" the houses that had the blood of a lamb on the door posts. Jesus later became "the passover Lamb". Easter is supposed to be The Passover, but they keep changing the dates, and it has become more of a pagan holiday these days (chocolate eggs, etc).
Some people will say that God uses His calendar for certain prophetic events, e.g. the Latter Rain could fall on the same day as Pentecost. That's interesting, and is the exact reason why I keep an eye on the feast days. But at the same time I must say that the Latter Rain is probably a much more finetuned event which will most likely fall on another day.
If you want to keep the feasts that is up to you. The important thing is that we don't judge each other for it.
The Bible also teaches us that we should not let others judge us for it, e.g. it's okay to stand up for it. There's nothing wrong with keeping the Sabbath for example.
Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days: Which are a shadow of things to come; but the body is of Christ.
- Colossians 2:16-17 (KJV)
I found this to be a very interesting topic, and I encourage you to look into it for yourself.