I have two finished articles (each about 25 pages long) but the second one uses results from the first one, none of which has been published yet. I would like to send them to some standard journal for consideration but I do not know exactly how to do so. Should I put them up on arxiv, refer back to the first article for the relevent results and send them to the respective journals? Or should I send one at a time? I do not know if this is the right place to ask this question, but I really wanted to get some professional opinion.
Put both papers in the arxiv. Have the second cite the first, if it depends on the first. You may have the first cite the second as a "preprint" as a motivation, if it's useful. This can be updated by re-posting the first paper once you know the "coordinates" (arxiv number) of the second.
If the papers form a clear sequence, you may wish to submit both papers together to the same journal. Explain why you are submitting two papers and their relation to each other in the cover letter. Otherwise submit them separately.
This situation frequently happens. In the era before the arxiv, I usually submitted such a pair of papers in one journal (simultaneously) and indicated which should be published first. In the second paper I would make the reference on the first one (calling it a preprint). The purpose of this is that very likely they will go to the same referee.
Arxiv makes the things simpler, because a reference to arxiv is a legitimate reference, and this gives you flexibility to send them to different journals, if you prefer. This may be a reasonable option if you want them to be published as soon as possible. Some journals are reluctant to publish two papers of the same author in a short sequence.
Posting all your papers on arxive has a lot of advantages. One of them is that you secure your priority, can put them on your vita and grant reports, and don't have to care how long will the journal publication take.