There is a short and important answer to the original question; then there is a longer and less important answer; the latter might, somewhat unfairly, prove more interesting.
The short answer: I do not believe in God through any sort of heroic effort of my own; rather, I believe in Him because He, through His grace, instils that faith in me. Faith is one of the three theological virtues (along with hope and charity) which we can't increase through any work of our own, but which is infused by God. But we can resist.
So why do I (try) not (to) resist?
This is the longer answer, and I can't hope to begin to do it justice here. I am not at all qualified to do so with the sole exception of the fact that I happen to be sitting here at the moment. I'll cover things more or less as they come into my head.
Miracles
Certainly not the best reasons, and no use as a foundation for faith, but very helpful for giving us a kick up the posterior, giving lie to our naturalist-materialist-nihilist philosophies and shoving us in the right direction. Every canonised saint has two miracles attributed to their intercession after their death, at least since whenever the regulations were last changed; these are often of the sudden-cure-of-terminal-disease sort. Some Wikipedia articles on more recent and dramatic miracles:
Our Lady of Fátima
Padre Pio
Our Lady of Lourdes
We can always dismiss miracles, and in fact we are right to be sceptical, since for every genuine one (begging your indulgence), there are countless mistakes, frauds, or indeed deceptions of the Enemy. But we can't discount them
a priori.
Right, I'm running out of time now, so what follows will just be sketches.
Sense of the numinous
Sometimes we have an indescribable sensation of the divine. Maybe we see the lives of other people who believe in God and recognise something utterly foreign to our own
but for some reason we can't quite place, eminently good; or we experience something beautiful. Now left undirected, this can lead us wildly astray. I mean, Palestrina's
Missa Papae Marcelli is beautiful, and Bach's
Fugue in A Minor is beatiful, but Palestrina was Catholic and Bach was Lutheran, so they can't
both be right. But these sorts of things are beginnings, not ends. They bring us to the first realisation that we are not our own gods, that we should not be burning incense on the altars of our own thoroughly fallible and capricious reason; and God does the rest. On the subject of beautiful things, I particularly like the
introit for the feast of the Immaculate Conception.
The lives of the saints
There are lots of people who lived and died before us who believed in God. Many of them did extraordinary things; so many martyrs could have saved their earthly lives by one small capitulation, one denial; but they chose rather to endure death at the hands of their persecutors out of love for their Creator.
Okay, that was ramshackle. Please point out the holes and I'll try to patch them up.
EDIT: Oh, afterthought, in response to earlier stuff. This is just for the record, and I'll talk more about it if anyone asks, but there is no selective use of Scripture in (authentic) Christianity; it's either all or nothing, and I contend that it ain't nothing. Also, the prohibition against judgement means we shouldn't presume to judge
subjectively whether a sin is committed, but it doesn't mean we can't say that stealing is wrong. Stealing is wrong, and someone who steals is objectively a thief, but maybe they're insane or so poorly formed that they don't realise what they're doing is wrong (which is very different from having convinced themselves that what they're doing is okay). But we still have the right, and indeed the obligation, to teach that stealing is wrong and to punish thieves. Condemn the sin, not the sinner: God will do the latter if he deserves it; we aren't able to tell in this life.