I’ll cover this question in two parts: the first is – “If I were to be left behind (on Judgment Day)…” 
     There are ‘two’ judgments (not per person, but two different times for different groups), because there are those of the ‘first resurrection’, who receive their ‘reward’ by Christ at His coming. Those are the ‘sheep‘ on the right hand of Matt.25. Then there are those of the second resurrection, those who come to Christ during His thousand year reign. They will not experience the ‘second death’, which is the casting into the ‘lake of fire’ at the end of that thousand year period. Thus their ‘reward’ in Christ comes at the end of the thousand years, after Satan is loosed again for a short time (see Rev.20 for all this).
     Firstly, the Judgment of God’s Great White Throne Judgment happens at the ‘end’ of Christ’s thousand year reign on earth (Rev.20:11). ‘Final’ Judgment does not happen with Christ’s return. The final Judgment will cast anything evil which is against God’s Kingdom into the ‘lake of fire’. That includes Satan, hell, i.e. the ‘pit’, his followers, etc.
     That idea there of being ‘left behind’ is a tradition of man not written in God’s Word (study 1 Cor.15; 2 Cor.5 about being ‘absent from the body’; Eccl.12 about what happens to each at death today, the flesh going back to the ‘elements’, and the ‘spirit’ going back to God).
     Read Zechariah 14 and Acts 1 together, to discover just ‘how’, and ‘where’ our Lord Christ returns. Discover from 1 Cor.15 just on which ‘trumpet’ our Lord Christ returns on (the “last trump”, or ‘trumpet’ Paul wrote in 1 Cor.15, which is the “seventh” angel sounding of Rev.11:15). The Left Behind Series books and movies are against God. The deception message taught in them is not from our Heavenly Father and His Christ. Fanciful reading though, hmm? How silly it is to put man’s books over God’s Word! And that’s exactly what many do when they have not thoroughly studied God’s Word for themselves, but instead seek to some popular tradition which the world creates.
     The judgment of appearing before Christ, which happens at our Lord and Savior’s return, is one of reward for His servants, but wrath to the ‘goats‘ put on the left hand (see Matt.25:31-34). But that is not the The White Throne Judgment Day of God which happens later, written of in Rev.20:11. Do not leave out the ‘thousand years’ Millennium of Christ’s reign, for it is clearly written of in God’s Word:

Rev 20:2     And he laid hold on the dragon, that old serpent, which is the Devil, and Satan, and bound him a thousand years,
3    And cast him into the bottomless pit, and shut him up, and set a seal upon him, that he should deceive the nations no more, till the thousand years should be fulfilled: and after that he must be loosed a little season.
4    And I saw thrones, and they sat upon them, and judgment was given unto them: and I saw the souls of them that were beheaded for the witness of Jesus, and for the word of God, and which had not worshipped the beast, neither his image, neither had received his mark upon their foreheads, or in their hands; and they lived and reigned with Christ a thousand years.
5    But the rest of the dead lived not again until the thousand years were finished. This is the first resurrection.
6    Blessed and holy is he that hath part in the first resurrection: on such the second death hath no power, but they shall be priests of God and of Christ, and shall reign with Him a thousand years.
7    And when the thousand years are expired, Satan shall be loosed out of his prison,

     At the end of Rev.20, after God’s White Throne Judgment, which is at the ‘end’ of Christ’s thousand year reign, that’s when anything evil, including the devil, his abode of hell, etc., will go into the ‘lake of fire’, and be no more. The burning forever idea is an ‘idiom’ for those rebellious who refuse our Righteous KING Christ Jesus. They will not be remembered any more; they’ll be merely ‘blotted out’ as if they had never existed. That’s when GOD’s Kingdom will be setup, and this old earth made ‘new’; refreshed is what the Hebrew and Greek meaning is (Psalms 37; 2 Peter 3:13; Rev.21:1; Isaiah 65:17; Isaiah 66:22).
     Our Lord’s teaching about the ‘sheep‘ being put on His right hand, and the ‘goats‘ being put on the left hand, means during Christ’s thousand year reign, there is going to be a separation between His, and those who refused Him today. We will all be in a ‘spiritual’ Kingdom under Christ Jesus as KING (see Rev.11:15; Rev.19:16; Rev.20 – 22). The unrighteous will not go into the ‘lake of fire’ yet, for God’s People are going to help our Lord Christ to do a little ‘boot camp’ drill, trying to get those unrighteous deceived straightened up. Satan being locked in the ‘pit’, will not be allowed to prevent the great teaching of God’s Word during that time (Rev.20:3). Isaiah 13-14 speaks of this ‘day of the Lord’, which is the beginning of Christ’s thousand year reign, but there is much ‘metaphor’ in those Chapters of Isaiah. I did some studies of them on my website, explaining many of the metaphors.
     God loves all His children He created, and doesn’t wish to lose any of us. That’s what Christ’s thousand year reign is for. All those who won’t, or can’t come to His Truth today, will know during that thousand years. God is most loving, and most fair. It is wrong to teach that the ‘lake of fire’ event happens when our Lord Christ returns, and that tradition is not according to His Word. We serve a Just GOD, most fair and long-suffering. None will perish until they have a ‘chance’ to hear the full ‘Wine’ of God’s Word, untainted, with ‘full’ understanding. Many today do not have that ‘chance’ of clearly understanding God’s Word for themselves. That’s what the thousand year reign of Christ is for. The last nine Chapters of Ezekiel reveal Christ’s thousand year reign of the future. Ezekiel 44 reveals that separation during Christ’s Millennium reign also. Those Ezekiel 40 – 48 Chapters link with Rev.22.

The second part of the question – “…what would I have to do to show God that I believe and that I want to serve Him?”
     To show God that we believe, and want to serve Him only;… get into His Word He sent to each one of us, learning how to do it His Way, and not the way ‘we’ think He wants. There’s only ONE WAY to be in God’s Will, and that is to know what He said do, and what He said not to do. It’s as simple as that. Those who don’t care to study It for themselves, at least trying,… well who do they have to blame? Themselves, for they don’t know what God wants them to do, for they haven’t read God’s Word! Part of the ‘big test’ is just opening His Word up for ourselves, and trying to understand It, asking His Guidance.

Isa 55:1   Ho, every one that thirsteth, come ye to the waters, and he that hath no money; come ye, buy, and eat; yea, come, buy wine and milk without money and without price.
2    Wherefore do ye spend money for that which is not bread? and your labour for that which satisfieth not? hearken diligently unto Me, and eat ye that which is good, and let your soul delight itself in fatness.
3    Incline your ear, and come unto Me: hear, and your soul shall live; and I will make an everlasting covenant with you, even the sure mercies of David.
4    Behold, I have given Him for a Witness to the people, a Leader and Commander to the people.
5    Behold, thou shalt call a nation that thou knowest not, and nations that knew not thee shall run unto thee because of the LORD thy God, and for the Holy One of Israel; for He hath glorified thee.
6    Seek ye the LORD while He may be found, call ye upon Him while He is near:

     Those verses are not just towards old Israel; they are for us today just as much; to all who call upon The Name of The LORD, of both Jew and Gentile (Romans 2:10).