Also what is the quickest and easiest way to get my upload ratio increased?

In addition to the freeleech approach mentioned by @bi4smooth above, you can download a torrent file into your client but only select a very few files to download (maybe 10% of the material). If you do this very soon after the torrent is posted to this site, you'll get that small amount of material very quickly, and then become a "source" for that material for other leechers, and you'll get a lot of upload credit.

Also, keep seeding anything you're already downloaded, even if you don't want to keep the material for your own collection. I have some torrents that I downloaded years ago that occasionally get a hit from someone.

Finally, if you have some material tat isn't already on this site, create a new torrent to upload, and that will get you a bunch of upload credit. If you don't have anything like that, I can put some on a cloud site that you can download from and then post here. Just send me a PM and we can work the mechanics off-line. (I hope this doesn't break any site rules.)

In order for my upload to start counting do I have to download the entire torrent file before it will start counting any of my upload?

No, uploaded data always counts - regardless of how much you have downloaded (now or in the future) - if you share, you get credit.

To my limited knowledge, the only "free" (un-charged download or un-credited upload) is the freeleech program, which allows you to download for free, although the people you download FROM still get credits for their sharing.

Because of this, one of the FASTEST ways to get "ration" help is do download a FREELEECH torrent when it has been freshly "free'd" - because there won't be many seeders yet, but there will be significant demand (there always is for freeleech content!) -- but if you download a freeleech after it has been free a while, there will be a LOT of seeders, and fewer downloaders - so of much less help.

Or will it still count my upload if I only download the parts I want and not the rest?

Yes, all uploaded bytes count - thought they don't show up instantly -- there is a scheduled "job" on the website that collects the data from the tracker (no, they are not the same systems - at least not as far as I can tell)

Also what is the quickest and easiest way to get my upload ratio increased?

See Freeleech description above, then keep an eye out for "fresh" freeleech torrents! (EVEN IF you don't want to KEEP the content itself!)