What are the Church's beliefs regarding carnal warfare and military service?
Key Verse:  He that leadeth into captivity shall go into captivity: he that killeth with the sword must be killed with the sword. Here is the patience and the faith of the saints.  [Revelation 13:10]

Are you thinking about entering Military Service?  Or do you know someone who is?  For a decision in your life as important as this, we should seek God's counsel.
Trust in the LORD with all thine heart; and lean not unto thine own understanding.  In all thy ways acknowledge him, and he shall direct thy paths.  Be not wise in thine own eyes: fear the LORD, and depart from evil.  [Proverbs 3:5-7]
God had commanded us not to kill our fellow man.
Thou shall not kill.  [Exodus 20:13]
In today's world, war seems inevitable, as more and more countries seek to flex their military muscle, which in most cases would be the expense of starvation of their people.  But if we all stop and think a bit, we would see that the result of warfare is taking the lives of other human beings.  The teaching's of Jesus shed light on this subject.
Then said Jesus unto him, Put up again thy sword into his place: for all they that take the sword shall perish with the sword.  [Matthew 26:52]
He that leadeth into captivity shall go into captivity: he that killeth with the sword must be killed with the sword. Here is the patience and the faith of the saints.  [Revelation 13:10]
In some countries, the citizens are bound by law to sign up for military service.  However, they have choices to serve in non-combat areas of the military, or even give voluntary service representing their country in another country.

While there is so much talk about right to life, there is the development of new and sophisticated weapons with the sole intent to kill or murder.  Children of God are forbidden to take another person's life Exodus 20:13.  Instead, we are to focus on the spiritual battle God asks us to fight.
For though we walk in the flesh, we do not war after the flesh:  (For the weapons of our warfare are not carnal, but mighty through God to the pulling down of strong holds;)  [2 Corinthians 10:3-4]
Avenging oneself isn't what God wants of his children.
Dearly beloved, avenge not yourselves, but rather give place unto wrath: for it is written, Vengeance is mine; I will repay, saith the Lord.  Therefore if thine enemy hunger, feed him; if he thirst, give him drink: for in so doing thou shalt heap coals of fire on his head.  Be not overcome of evil, but overcome evil with good.  [Romans 12:19-21]
It is our understanding and belief therefore that we should not participate in the act of war where we might have to kill another individual.  We are to focus on the battle of saving lives for Jesus.  The Armour we are expected to put on for this battle is found in Ephesians 6:11-18
Put on the whole armour of God, that ye may be able to stand against the wiles of the devil.  For we wrestle not against flesh and blood, but against principalities, against powers, against the rulers of the darkness of this world, against spiritual wickedness in high places.  Wherefore take unto you the whole armour of God, that ye may be able to withstand in the evil day, and having done all, to stand.  Stand therefore, having your loins girt about with truth, and having on the breastplate of righteousness;  And your feet shod with the preparation of the gospel of peace;  Above all, taking the shield of faith, wherewith ye shall be able to quench all the fiery darts of the wicked.  And take the helmet of salvation, and the sword of the Spirit, which is the word of God:  Praying always with all prayer and supplication in the Spirit, and watching thereunto with all perseverance and supplication for all saints;  [Ephesians 6:11-18]
This Armour helps to save lives rather than take lives.  If the soldiers of his world would be preachers of Truth everywhere, there would not be such a thing as warfare. Think on these things and may God guide the making of the right choice.