Old Testament
If you read the Old Testament, the rituals mentioned are pretty much about:
- Reconciling Israel with God
- That's what all of the sacrificing, avoiding unclean things, etc. is all about.
- Detailing the consequences of obedience & disobedience to God's laws
- Great example is when half of Israel stands on a really fertile hill and the other half on a really barren hill. The ones on the fertile hill call out all the promised blessings across the gap while the ones on the barren hill shout out the promised curses.
- Specifically prescribed one-time events
- like when the armies of Israel walked around the walls of Jericho for a week and the walls come crashing down, or
- when the Philistines come running up to raise Cain after Israel spent a bunch of time repenting and purifying themselves, and were terrified by a huge clap of thunder coming from a completely clear sky right as they reached their first target so they headed back down to their boats and took off.
Also while we frankly don't really know what happened in the Jewish Temple beyond the day of atonement stuff and chores in the hall of service, we do know that there were rites and ritual happening from other writings from the time that hint at them. We just don't know any details at all; they don't seem to be written down (or if they were they were lost some time after the Romans sacked Jerusalem. There was an active Jewish Temple being maintained in Egypt during and after that as well, but I don't know how long that lasted after the Romans took Israel as a province, and it seems to not have left details behind either.)
That being said, what isn't given are rituals that just anyone can perform. Any specific rituals given in the Old Testament are to be performed by the Levites under the guidance of the priests in the lineage of Aaron, the priests themselves, or sometimes only the High Priest (also descended from Aaron). It's also made pretty clear that anyone taking it upon themselves to usurp that privilege is making a mistake (see the man who decided during the 40 years in the wilderness that the ark was tipping a bit, and that it would be okay for him to reach out a hand and steady it even though it was expressly forbidden for anyone not a Levite or Priest to touch it; he died instantly.)
I would also add that many of the rituals in the Old Testament follow the pattern of other rituals being performed during that time period, in that it involved killing a lot of animals, which today is usually frowned upon by polite society.
New Testament
The New Testament is a different beast. Where the Old Testament is a combination of cosmogony, history, prophecies, and wisdom literature, the New Testament is made of first-hand accounts of the life of Christ, the account of his Apostles immediately after the crucifixion, letters from the Apostles, and the revelation of John to the seven churches.
There's one ritual described, when Christ instituted the symbolic eating of his flesh and blood in remembrance of Him, in lieu of the animal sacrifices that were so common at the time. This is one ritual that has since been clarified that it must be performed by an ordained priest.
Honestly, of all the potential rites/ritual of the Bible, the one that's easiest to attend and probably the most useful to someone is participating in the sacrament of the Lord's Supper. In most countries the odds of finding a Christian church to celebrate at on a Sunday are pretty high. (It's harder in Muslim countries--in many of them it's actually illegal to practice other faiths, so you'd be less likely to find a Christian church there.)
If you're looking for specific ritual to adapt for a magical practice, there's not really anything you can just take and use straight from the Bible. If you're looking for things you can do that incorporate Biblical writings however, there are many groups that use the Psalms that aren't particularly Christian (Rootworking, Hoodoo, etc), as well as Christian groups that use stories from the life of Christ or other prophets as part of their rites & rituals.