I am really new to this. If I install Ubuntu 17.10, which is supported for the next 9 months, then if there's an update to Ubuntu (in this case 18.04 LTS) will it auto update to that version? Otherwise after 9 months would I need to get the newer version's ISO on the Ubuntu desktop download page?
Although your Ubuntu system will not automatically upgrade itself to the next release of Ubuntu, the Software Updater will automatically offer you the opportunity to do so, and it will also automate the process of upgrading to the next release.
There is potential for confusion here because there are two things someone might mean when they say the release upgrade is automatic. One possible meaning is that it will happen even with no action from the user, which is not the case. Regardless of the user's configuration, no Ubuntu release will start an upgrade to a later release without the user's action.
However, Ubuntu will automatically tell you when a new release is available and as you if you want to upgrade to it. Even if you say no, you will still be able to go into the Software Updater and start the upgrade to the newer release.
This is different from security and stability updates within a release. You should almost always install those immediately when they become available, and you can configure Ubuntu so that this can be done behind the scenes without even prompting you. In contrast, unless you write a script yourself for this--which I strongly recommend against--you will never sit down at your computer to discover that you are, all of a sudden, running a different release of Ubuntu than you expected.
Although some users prefer to reinstall from scratch for each release, you don't have to do this. Only if you want to do that will you need to download the ISO image for the next Ubuntu release. The Software Updater does not need this ISO image to perform a release upgrade. It will download everything you need automatically after you tell it that you do want to perform the release upgrade.
Now, with all that said:
You might want to have an ISO image for the release of Ubuntu you are running or plan to upgrade to on hand, because you can use it as a live environment in which to run Ubuntu. This has two advantages: it should help you test if that release will be able to use your hardware successfully without any additional configuration, and you can use it if you ever need to run that release of Ubuntu but you don't have a computer with it installed and working. Note that these advantages apply regardless of how you upgrade, because they are separate from upgrading.
You might want to install an LTS release, currently 16.04 LTS, instead of a non-LTS release like 17.10. LTS releases are supported longer--LTS stands for "Long Term Support"--and they are almost always more stable than non-LTS releases. When you have an LTS release, you can still upgrade to a non-LTS release if you so choose, or you can skip them and (eventually) upgrade to the next non-LTS release.
The disadvantage of using an LTS release when there is already a newer non-LTS release is that the LTS release doesn't have the newest software and features. But that is, well, also the advantage of LTS: the software you use is more tested and established. Most users will probably prefer to stick with LTS releases, but even non-LTS releases are heavily tested and you should definitely feel free to use them.
You may also be interested in what it is like to upgrade from one Ubuntu release to another. For that, I recommend you take a look at this official walkthrough. We also have How do I upgrade to a newer version of Ubuntu?, which is useful but might make upgrading seem more complicated than it usually is because the goal of its answers is to cover most of what one might ever have to consider or fix.
In a few words, it will "auto update". It does depend how you've set up the "updates" section. Go to System Settings, Software / Updates and check the Updates tab to see how it set up.There are a number of "update" options there, so choose what you think is best for you. My preference would be, if this is a fresh install you've done, check it and leave it.