This is one of those questions that should have a clear, easy answer, but when you ask "what do Christians believe about this" you will likely get a lot of different answers. I'm going to put a preface explaining why those answers vary, and then go into a purely Scriptural view, which is the view you will likely hear from the pulpit of any Church you visit.
Why the answers may vary.
In Christianity, there is a concept of being "free from the Law of Moses" through the perfect substitutionary sacrifice of Christ. The central theme of Christianity is that we can never be "good enough" to get to Heaven on our own, and that Jesus is the promised Christ - His sacrifice is what gets us to Heaven, not our own good works.
One of the questions that varying groups differ (and they differ wildly) is what, exactly it means to be free from the Law of Moses. There are other questions already on this site that explore that topic, but in a nutshell, there is debate over whether we are obligated to tithe, to refrain from certain things, to perform certain ceremonies, etc.
This applies to your question directly because some groups believe that we are under no obligation to obey the Old Testament commandments regarding this, and others will say we are. Some will tell you that the Law of Love dictates that we will meet or exceed the requirements of Mosaic Law, and some would accuse those people of Legalism.
One common teaching
The teaching with which I am the most familiar with comes from the mouth of Jesus, as recorded in Matthew 5:20-24 (King James version)
20 For I say unto you, That except your righteousness shall exceed the
righteousness of the scribes and Pharisees, ye shall in no case enter
into the kingdom of heaven.
21 Ye have heard that it was said of them of old time, Thou shalt not
kill; and whosoever shall kill shall be in danger of the judgment:
22 But I say unto you, That whosoever is angry with his brother
without a cause shall be in danger of the judgment: and whosoever
shall say to his brother, Raca, shall be in danger of the council: but
whosoever shall say, Thou fool, shall be in danger of hell fire.
23 Therefore if thou bring thy gift to the altar, and there
rememberest that thy brother hath ought against thee;
24 Leave there thy gift before the altar, and go thy way; first be
reconciled to thy brother, and then come and offer thy gift.
This is commonly taught as doing whatever it takes to make it right. If you have sinned against someone, or even if that person thinks you have sinned against them, and you hadn't intended to, the priority is to reconcile and heal the relationship.
Per His words in verse 24. reconciliation with an offended person takes priority over worship. (How can we come to God with a clean heart, if we are carrying the stain of sin?)
Clarke's Commentary on the Bible expands on this:
Leave there thy gift before the altar - This is as much as to say, "Do
not attempt to bring any offering to God while thou art in a spirit of
enmity against any person; or hast any difference with thy neighbor,
which thou hast not used thy diligence to get adjusted." It is our
duty and interest, both to bring our gift, and offer it too; but God
will not accept of any act of religious worship from us, while any
enmity subsists in our hearts towards any soul of man; or while any
subsists in our neighbor's heart towards us, which we have not used
the proper means to remove. A religion, the very essence of which is
love, cannot suffer at its altars a heart that is revengeful and
uncharitable, or which does not use its utmost endeavors to revive
love in the heart of another. The original word, δωρον, which we
translate gift, is used by the rabbins in Hebrew letters דורון doron,
which signifies not only a gift, but a sacrifice offered to God. See
several proofs in Schoettgen.
The answer to most of your sub-questions are simple, if we take His words at face value:
Is seeking forgiveness from God sufficient, or do you first need to
set things right with the other person?
We need to first make it right with the other person.
If the latter, what are you obligated to do -- apologize, make amends
(what kinds), something else?
Whatever it takes to reconcile. This may be as simple as an honest confession of your sin to the other person, and a sincere apology, it may involve paying something back, it may mean doing something you really don't want to do (so long as that thing isn't sinful). You may have to apologize before the Church, or admit to peers/friends/families that you've done wrong.
There may be cases where the other offended party will not accept any offer of reconciliation, at which case, we can only do our best. God knows our hearts and minds better than we can, and He knows both your mind and that of the person who holds something against you.
Does the answer depend on who was wronged? Is the obligation to a
fellow Christian different than one to a non-Christian?
No. Absolutely not. The obligation to a fellow Christian is no more or less than to a non-Christian.
I've taken the liberty of finding a few teachings on the subject: