WordPress is a great CMS for newbies due to its user-friendly interface and general intuitiveness, however there are some common problems that I've seen asked about time and time again in my capacity as Internet Forum Fiend.
The Ultimate WordPress Fix
Almost any WordPress fix starts with narrowing down the problem, and the first place to start is disabling all of your plugins. Plugins are created by, well, whoever decides they want to create a plugin, and as a result can sometimes be badly coded or just plain go out of date.
If turning off your plugins fixes your problem, start to enable them one by one until you narrow down which is the problem. You can also try switching back to the default theme in case the problem is with your theme's code.
A note on backing up
Worst case scenario, depending on your specific WordPress problem, you may end up needing to back up a previous copy of your site. A plugin like WP-DB-Backup will make creating regular backups a lot easier, so take advantage of it.
WordPress problem 1 - Disappearing posts
So you've visited your site only to find that the content has all disappeared? Step 1 is not to panic.
Try implementing the The Ultimate WordPress Fix, and if this doesn't work it's most likely to be a database issue. If your host is anywhere close to decent, you'll have access to a cPanel or other control panel. Find your way into "phpmyadmin" which is the interface for dealing with databases.
You'll need to figure out which database table is causing your problem. Usually it will be wp_posts, just check the box next to it and choose "repair" from the dropdown at the bottom of the list.
That should bring your posts back, if not it might be time to restore a backup.
WordPress problem 2 - "Broken" admin panel
This is one of the problems that can often be related to the theme, as well as the plugins, so be thorough and disable pretty much everything.
If none of that works it's possible you have corrupted files, this can sometimes happen when WordPress is uploaded to the server or upgraded. Try re-uploading everything in the wp-admin/js folder.
WordPress problem 3 - My images aren't showing up
This is less likely to be resolved via The Ultimate WordPress Fix, but you never know so try it before doing anything more drastic.
This problem usually involves images on your homepage, or in a slider of some kind. A lot of themes nowadays take advantage of featured images with each post and use a script called "timthumb" to resize them for various locations on the site.
Step 1 is to find out exactly how the images are being included in your page, since that will obviously affect which is the appropriate fix.
While on the broken page, view source to see the HTML. This is usually ctrl-u depending on your browser. What you need to do is find the code where the image should be showing up. If you're not sure what to look for an easy trick is to find some text near the image on your page, and then search for that text in the code, hopefully you'll see an <img> tag somewhere close and this is your broken image.
Try copying everything in the image source, paste that into your address bar in your browser and see what happens. If an image shows up, then I'm as stumped as you are.
If the source involves timthumb.php anywhere in it, try and follow along with this example:
The image source is currently:
and no image shows up when you visit this URL directly. Try changing it to:
Note that http://thingy.com/ has been removed from the image path at the end.
If removing the domain works, we've found your problem. Either remove the domain when you add an image via custom field, or if this isn't how your theme works you can edit the actual theme files (or pay someone to do it for you).
Note: If your theme is not using timthumb or any similar resizing script, and is instead using WP to resize the images, there's not much that can be going wrong. Check your media library to be sure the image hasn't been accidentally deleted.
WordPress problem 4 - There are weird links on my page
Before you do anything else, try The Ultimate WordPress Fix just in case.
Now answer this question honestly: are you using a free theme? If so, they often come with non-removable spammy links or even worse, are actually dodgy copies of premium themes that you could be asked to remove if found out.
The first thing to check is your theme's footer.php file. If you can see the link text in there, or any weird strings of seemingly random numbers and letters, then these are coded into your theme and it may be against the license terms to remove them. Remember, someone worked to create this theme, going against the license is the same as stealing.
If you see nothing in your footer.php, I have bad news for you. The linnk code could actually be in any file at all, I just started you off in footer.php since that's the most common place to find them. Check header.php next, and work your way through until you're sure the links aren't hardcoded in.
If the links are definitely not generated by your theme, it's possible you've been hacked. You may need to get in the professionals, here.
Note: Many free themes include link code that is not actually visible on the page. It's always a good idea to check free themes for any dodgy code, starting with the footer.php. You may also gain clues from the license file that should have come with the theme, and whether or not it mentions ads/credit links.