If you are able to produce notes, but they don't sound right – trust what your ear is telling you because your ear, unfortunately, is probably correct. It can happen if you accidentally play the wrong note (because you misread the music or tablature), or if you play the correct note in a manner than produces a sound of poor quality. If you produce a note that sounds bad, freeze both hands as soon as it happens and look at them carefully, to see if any of these could be causing your problem:
1) First make sure that you are playing the right note – meaning, verify that the fret you played matches either the note in the music, or the notation in the tablature. If you find that you did make a mistake, then back up and play that entire measure again. (If the note you missed is the first note in the measure, then play the measure immediately before where you made the mistake, and then the measure where you did make the mistake. Practice ONLY that short sequence, over and over again. It takes a lot of effort to "unlearn" something that you have "learned" incorrectly, and the general rule of thumb is that, when you "learn" a mistake, you have to be able to play the notes correctly seven times in a row, in order to "unlearn" your mistake. And, as if things are not difficult enough when you’re just starting out, I want to be fair and tell you that you are likely to find a fair amount of mistakes in written tablature. Written music is generally much more accurate. But if you are positive that you are following the tablature correctly, and you're privately thinking that it can't possibly be right – well, you could be correct about that. If your ears tell you that something doesn't sound right, trust them. But before you change the notation, get a second opinion from a more experienced player, who will either verify that you did find a mistake, or who may be able to point out a nuance of the music with which you are simply not familiar. Many people think that Celtic music and jazz "sound wrong" even when the written notation is correct, because both can have a moody and disturbing quality.
2) Are the tips, the only part of your finger that is touching the string? Some musicians tend to flatten their fingers out when they play. In order to produce the best quality sound, your fingers need to be arched so that only the very tips press on the fingerboard, and only on the string that you're playing. That means that you need to keep your fingernails very short – otherwise your fingernails interfere with your ability to press the strings to the fingerboard.
Sometimes musicians flatten their fingers out when they’re trying to reach over to the "G" string. I think they do this because they think that their left hand isn't allowed to move, which is not the case Assuming that you have a mandolin strap, your left hand should not be needed to "hold" your instrument. (If you don't have a mandolin strap, and you are having this problem, this might be a good reason to get one.) But when your strap supports the neck of your instrument, your left hand has the freedom to roll over the top of the fingerboard a bit, so that you can reach the G string quite easily without having to flatten your fingers.
3) Check to see where your finger is, in relation to the fret. While it's true that pressing your finger anywhere between two frets will produce a reasonably accurate note, you will produce the best quality note if you press directly behind the fret. If you press directly on the fret, it sounds really awful.
4) Are you pressing the string hard enough to get it down to the fingerboard? You will not produce the correct note unless you do so, and the question does require a serious evaluation of how you’re playing, in order to find the appropriate fix. First, are you having trouble pressing all three fingers all the way to the fingerboard, or just one? If you're having trouble pressing all three, then you may want to take your instrument to a professional to have your action lowered. "Action" refers to the distance between the strings and the fingerboard. Some musicians prefer high action, and some musicians prefer low action. But if you can't press your strings all the way down to your fingerboard without a struggle, then your action may be too high for you. And that can be fixed with a simple adjustment. If you’re only having problems pressing one finger all the way, then you will probably want to leave the action alone, and just practice more with that finger until it develops enough strength to press the string all the way down. Having the action too low for the other two fingers can cause a whole different set of problems.