What happened after we were buried with Christ—did we stay in the tomb?

No, we did not remain in the tomb. When Christ was buried, we were buried with Him in the mind of God, but we did not stay there any more than He did. We were resurrected out and given new life, eternal life, to live forever in His presence.

Les explains that chapter 6 teaches that when Christ died, we died; when He was buried, we were buried, in the mind of God. But we did not stay in the tomb any more than He did. We were resurrected out, and we too were given new life. Les points ahead to chapter 8, where the word life means eternal life, and believers will live forever in His presence. He contrasts that with even a 90-year lifetime, which is nothing compared to the billions upon billions of years of eternity, and says this book prepares us for that coming eternity.

Romans 6:4

Therefore we are buried with him by baptism into death: that like as Christ was raised up from the dead by the glory of the Father, even so we also should walk in newness of life.

Romans 6:5

For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection:

Are we buried with Christ by water baptism or by God's determination?

Believers are buried with Christ not by water baptism but by God’s own determination. God saw every believer on the cross and in the tomb in the person of Christ, reckoning them crucified, buried, and raised with Him to new life. It is a God thing, accomplished by His power.