According to Baptist teachings, what are the actual precepts God requires of us who are his? I am reading Psalm 103 and I come to the word "precepts" in these verses in the NIV:


  17 But from everlasting to everlasting the Lord's love is with those
  who fear him, and his righteousness with their children's children—
  18 with those who keep his covenant and remember to obey his precepts.


According to Baptist teachings, what precepts does God require from his children, particularly in reference to this Psalm?

Lord's Blessings.
It is pretty widely accepted, that the old testament Jews believed this too be the most important precept,  was this
Deuteronomoy 6:4-5 called the "Shema"


  4 “Hear, O Israel: The Lord our God, the Lord is one![a] 5 You shall love the Lord your God with all your heart, with all your soul, and with all your strength.


The next section is good too. Deut 6:5-9


  6 “And these words which I command you today shall be in your heart. 7 You shall teach them diligently to your children, and shall talk of them when you sit in your house, when you walk by the way, when you lie down, and when you rise up. 8 You shall bind them as a sign on your hand, and they shall be as frontlets between your eyes. 9 You shall write them on the doorposts of your house and on your gates.


The idea being, that your life is devoted to obeying the Lord. You write his commandments on your front door, you talk about them all day, at home, with your friends, your family, you meditate on the word as you wake up, and before you go to sleep.

Compare with Proverb, 3:6


  In all your ways acknowledge Him,
  And He shall direct[a] your paths.


Making a list of all the "precepts" of God for human life could take a very long time. God understands the variety of human life, and in the Bible has left 66 books, covering a variety of issues, times, and situations.

The bible speaks on every facet of human life, God has things to say on, How you treat your family, How do you share your faith, how to be a good husband, how to be a good wife, how to obey the authorities in your life who are christian, how to obey authorities in your life who are secular, principles for giving money, principles for sexual behavior, principles for morality, principles for treating other believers, principles for running a local church etc etc etc.

A list of all the different precepts would be extremely long. When Jesus was asked a similar question, here is how he responded.
Matthew 22:35-40


  5 Then one of them, a lawyer, asked Him a question, testing Him, and saying, 36 “Teacher, which is the great commandment in the law?”
  37 Jesus said to him, “‘You shall love the Lord your God with all your heart, with all your soul, and with all your mind.’[d] 38 This is the first and great commandment. 39 And the second is like it: ‘You shall love your neighbor as yourself.’[e] 40 On these two commandments hang all the Law and the Prophets.”


Many times, people want to look for a few specific rules which they can keep perfectly, and then they will be able to feel like "Good" people.

The bible is clear that we are not saved by obeying the law, rather we are saved by Grace, and any "good works" are a result of being saved, and not what causes us to be saved.

see Ephesians 2:8-10


  8 For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, 9 not of works, lest anyone should boast. 10 For we are His workmanship, created in Christ Jesus for good works, which God prepared beforehand that we should walk in them.


So with all this being said, God certainly does have standards for his children, there are absolutely "do's and dont's"

The classic example of these would be the Ten Commandments.  You can find these in Exodus 20.

These lay out foundational principles for how we are to honor God, and how we are to honor other human beings as well.

*footnote
Your question was addressed to "Us who are His". This is who these are addressed to. These are built on main requirement of God towards people
Jesus said in Mark 1:15


  14 Now after John was put in prison, Jesus came to Galilee, preaching the gospel of the kingdom[e] of God, 15 and saying, “The time is fulfilled, and the kingdom of God is at hand. Repent, and believe in the gospel.”