Surely when we guard our eyes and forego self-destructive behaviours we're really serving ourselves rather than God right?
Well... not necessarily. There are many commandments that we keep, even though they are not self destructive, in order to serve hashem. For example, masturbation. It is arguably a "healthy" behaviour but we try to limit ourselves because hashem commands us not to. P in the other hand is arguably a self destructive behaviour but we try to limit ourselves because that is God's will. So we are serving ourselves BUT our main incentive to stop is because it is hashem's will (the proof to this is that most secular people don't think twice about it even though it is morally questionable while we are here struggling).
Also, what does it mean to have a connection with God? Does this mean spiritual sensitivity and appreciation of beauty in music, nature and literature or am I totally wrong?
It means different things to different people. As Holy Yid/Yiddle2 wrote, there are many different approaches to this but it definitely can mean spiritual sensitivity and appreciation of the beauty in this word.