I am working towards establishing my small design business. I will be making primarily stationery (notepads, letter paper, shopping list pads, post-it type of note paper, etc, perhaps also greeting cards in the future). I will be using my own images/illustrations for these (I took photos myself and created illustrations too), but occasionally I would like to include a quote or just a small phrase in some of my designs. I could probably create my own font(s) eventually, just right now I have no time for that. So, I'm thinking about purchasing a font or two to use commercially.

I'm wondering if I have a legal right to offer/sell my ready design(s) to publishers (who will print them out and distribute through their own catalogs or other channels), and part of these designs would be the font I purchased?

I do not intend to send a client/buyer the font as a separate file. It's not like I'd re-sell the font itself, it would simply be included as a part of the entire image, maybe even not all alphabet letter would be used at once. It could even be just one letter (like the first letter of a person's name, for example).

Also, one more question: Are there actually fonts that are completely free, even for commercial use like in my case (re-selling them as part of a design)? Could anyone please direct me to them? I did a search online but am confused about most of the fonts I've found. I just want to do everything legally right.

The fonts are protected from unauthorized duplication because they are computer software. Note carefully that this applies only to the font software itself (the .otf or .ttf file), not the actual artistic design.
The simplest standard of legal font use is that no one is allowed to use a font on their computer unless they specifically have a license for that font.

Thanks, claudeserieux. Sorry, I may sound a bit dumb, it's just so new and complicated to me... So, does it mean I can use the font any way I want as long as I purchase it to be added to my computer? If I add the font to Photoshop or Illustrator and then sell my design to a publisher who will then resell it, am I still safe legally? Like, if I create a greeting card with my drawing and, say, a "Happy Birthday!" written with the font I purchased, I can offer this to a card publisher for profit?

Yes.
Some fonts may be embedded within document files. Embedding allows fonts to travel with documents. Embedded fonts can only be used to print, preview and in some cases edit the document in which they are embedded.

Thanks so much, claudeserieux! It's a great news. So, off shopping for fonts! :-)

personally i would just drop the font designer a message, as dafont states that "The licence mentioned above the download button is just an indication. Please look at the readme-files in the zips or check the indicated author's website for details, and contact him/her if in doubt. If no author/licence is indicated that's because we don't have information, that doesn't mean it's free."

Dead-Duckling, thanks. Good point. I personally wouldn't want to use free fonts because if the author in the future decided to start selling the font, I wouldn't want to get rid of it. Or keep an eye on the changes/updates, etc. It's always safer to buy it and have a proof of purchase, I guess. But I'll definitely check with the author.