Do we have to obey the laws of the bible? If so, what laws? I've read a ton about obedience in the bible and I realize that we have a responsibility to obey Jesus Christ. As it says in 2 Timothy 2:19 "God's foundation stands firm, sealed with his inscription. Anybody that professes the name of Christ must repent of their wickedness"

There's really no way around this. However, when I begin to speak about our responsibility to obey Jesus people begin to tell me that I am trying to follow the law or work for my salvation.

I know that neither of these will earn my way into God's grace, however, I still understand that I have a responsibility.

Where do you draw the line? What do we have to obey and what do we not have to obey?
Yes - but which of them is a matter of debate

Since I already did a lot of work to answer To what extent does the Law of Moses still apply? I'm going to adapt that answer here, too. Similarly, I'll list and briefly describe some of the more common views along with sources.

Catholic

The Catholic Church teaches that the Law of Moses (the Old Law) is a preparation for the Gospel, and as such no longer binding. The New Law (the Law of Gospel) is a perfection of it, given through faith in Christ.

The Law of the Gospel, which applies to Christians, is presented in the Sermon of the Mount (Matthew 5-7), but also in the moral catechesis of the apostolic teachings, such as Romans 12-15, 1 Corinthians 12-13, Colossians 3-4, Ephesians 4-5, etc.

God's Law has always remained the same, and has always been available as the natural moral law (e.g. through conscience), but only Jesus could express it perfectly.

Source: Catechism of the Catholic Church, Second edition. Paragraphs 577-582 and 1950-1986. http://www.scborromeo.org/ccc/ccc_toc2.htm

Reformed

The reformed teaching is that the law was given in three parts:


The moral law, which God gave first to Adam, and later to Moses on Sinai (the Ten Commandments).
Ceremonial laws, given to the people of Israel, prefiguring Christ.
Judicial laws, given to the State of Israel.


The moral law is eternal, binding Adam, the first man, as well as any Christian today.  Ceremonial laws were abolished in the New Testament. Judicial laws only concerned the State of Israel.

As noted, the moral law was delivered to Moses as the Ten Commandments. Thus the Ten Commandments apply to Christians today as such. Of other laws in the OT, the moral component applies. Moral laws given in the New Testament are also part of the eternal moral law, and meant for Christians of today.

Source: Westminster Confession, chapter XIX. http://www.reformed.org/documents/wcf_with_proofs/ch_XIX.html

Theonomy

Theonomy literally means "God's law". It is the notion that God's law is eternal and universally binding. That is, the entire Bible applies to Christians today. Every law God has given us is meant to be obeyed.

This doesn't mean that salvation comes by keeping the law, as theonomy is often misrepresented. Salvation is granted "solely by grace through faith in the person and work of Jesus Christ".

Psalm 119:97-98 (KJV) is a good verse to show what theonomy is about:


  O how love I thy law! it is my meditation all the day. 
  Thou through thy commandments hast made me wiser than mine enemies: for they are ever with me.


Source: Duncan, T.M. Theonomy: What it is; what it is not. http://www.ipc.faithweb.com/documents/THEONOMY.htm

Dispensationalism

Dispensationalism makes a distinction between Israel and the Christian church in God's plan. Its major objective is to be able to interpret the Bible consistently literally.

Dispensationalists recognize seven dispensations in the Bible:


Innocence (Genesis 1:1–3:7)
Conscience (Genesis 3:8–8:22)
Human government (Genesis 9:1–11:32)
Promise (Genesis 12:1–Exodus 19:25)
Law (Exodus 20:1–Acts 2:4)
Grace (Acts 2:4–Revelation 20:3)
The millennial kingdom (Revelation 20:4-6)


The different dispensations are different ways that God has related to people. Salvation has always been through faith.

As today's Christians are living during the dispensation of grace, or during the New Covenant, the old laws or the Old Covenant do not bind them. Because God and His will do not change, the moral law of the New Covenant hasn't notably changed from the Old Covenant.

The Old Testament thus is not for Christians. Laws expressed in the epistles are part of the dispensation of grace, and thus apply to Christians. Laws stated by Jesus are not, and thus are not really meant for Christians but for Jews. (Though I expect there to be a small minority of dispensationalists saying otherwise.)

Source: What is dispensationalism and is it biblical? http://www.gotquestions.org/dispensationalism.html (I'd like to find a better/more original source)