What else CAN and CANNOT a Christian do on the Sabbath in order to perfectly keep it today?"
You keep the Sabbath holy by not working on it, and resting. God said to rest on the Sabbath. However, sometimes we have to take into account what kind of job you have. If you are a doctor, and you HAVE to work on Sundays, God knows what you have to do. But the doctor should NOT work seven days a week. He should take a day off for rest as God commanded. (If someone gets in an accident on Sunday, someone has to perform surgery, or prescibe medication to save his life). There are emergencies that might come up as well. God wants us to deal with them, so we can have a good life, ignoring a crucial emergency on the Sabbath is not God's will. Take care of it as soon as you can then continue to rest(on the Sabbath).
8For the Son of man is Lord even of the sabbath day.
9And when he was departed thence, he went into their synagogue:
10And, behold, there was a man which had his hand withered. And they asked him, saying, Is it lawful to heal on the sabbath days? that they might accuse him.
11And he said unto them, What man shall there be among you, that shall have one sheep, and if it fall into a pit on the sabbath day, will he not lay hold on it, and lift it out?
12How much then is a man better than a sheep? Wherefore it is lawful to do well on the sabbath days.
13Then saith he to the man, Stretch forth thine hand. And he stretched it forth; and it was restored whole, like as the other.
Jesus healed on the Sabbath b/c there were pressing issues(the leprosy man). The Pharisees would have chose not to do ANYTHING, but God gave us the levity to perform some duties, even on the Sabbath for our good health and good welfare. (If your child was abducted on Saturday, you are not expected to stop searching for him/her on Sunday. No, that's a pressing issue, and Jesus gives us room, or levity, for that.
Sunday is the Sabbath. I think Seventh Day Adventists have Sabbath on Saturday, which some argue is the real Sabbath, but Christians observe the Sabbath on Sunday.
Remember, God knows our hearts, so if we sincerely desire to obey Him and His Word He will guide in the knowledge of Him, and what He wants us to do.