Although this is an excellent question, it is also such a large topic that libraries could be filled with the books that have already been written on this subject. I am sure you will understand that I can only touch the hem of the garment so to speak.
Several Bible passages immediately come to mind. I would recommend that you meditate on Romans 12:1-2 and 2 Corinthians 5:14-15. They have much that can help you understand the level of surrender the Lord expects of us. As Paul said in Romans 12:1, it is only our "reasonable service." However, for the sake of simplicity, I want to look at one particular verse.
Romans 6:13 Neither yield ye your members as instruments of unrighteousness unto sin: but yield yourselves unto God, as those that are alive from the dead, and your members as instruments of righteousness unto God.
Romans 6:13 gives us three steps for giving our all to the Lord. If we will study and meditate on these three steps, I think we can begin to understand what it means to give our all to the Lord. Here are the steps:
Refuse to give yourself to sin. The verse says that we are not to yield our members as instruments of unrighteousness unto sin. Your members are things like your eyes, your ears, your mouth, your mind, your hands, and your feet. In other words, take each part of your body and determine that it is not going to give in to sin. Refuse to let your eyes look on that which is displeasing to the Lord, to let your ears listen to anything they should not, to allow your feet to take you where you should not go, and so on. This step is negative, but it is the first step in giving yourself to God. You must first refuse to give yourself to your own desires and lusts.
- Yield your whole self unto God. To yield means to give in to, to submit, to surrender. Romans 12:1 speaks of presenting our bodies as a living sacrifice unto God. I believe this means that we should come to God in prayer and tell Him that we want to give ourselves unto Him. We choose to give Him our lives and surrender to Him by telling Him that we have decided to do so. You must decide that you want God to have you entirely and that you are willing for Him to be in charge of your life. Tell Him that you want Him to have you entirely. If there is any area of your life that you have not surrendered to Him, tell Him that it also is His to do with as He pleases. If you cannot tell Him that, then you are not ready for giving your all to the Lord.
Yield your members as instruments of righteousness unto God. This is the opposite of the first step. In the first step, you consider the different members of your body and refuse to give in to the desires of the flesh. Now, you must go back to each of the members and yield them to the service of God. You can practice this by praying like this: "Lord, where do you want my feet to go today? What do you want my mouth to say? Am I thinking about what you want me to think? Am I watching and listening to those things that are pleasing to you? Am I doing with my hands your will?" God will begin to lead you in His ways. As you grow in the Lord and are ready for more, He will reveal more and more of His will to you.
I know that I could tell you many of the things you should be doing, like praying and Bible reading and going to church and more. I am not discounting these at all. However, I believe this verse outlines how we can give our all to the Lord. You may fail on many occasions. In fact, if you follow these steps in sincerity, you may be more frustrated for a time because these steps will show you many ways in which you fail the Lord. You may not know these shortcomings at this time because you have not made the commitment and followed through with it. However, this path is the way of greatest blessing. When you fail, get back up and start all over again. Be obedient to the Lord and He will certainly bless your life in ways far beyond your understanding.