Others have already provided good (and bad) reasons to write concise proofs, but since you are planning to publish yourself, I will share my own approach when writing papers.
As an author, you have to satisfy very different types of readers, from undergraduate students to highly experienced researchers that work exactly in your field of research. But even for a single person, different levels of abstractions are necessary, because I (and surely many others) read papers top-down:
When reading a paper for the first time, I skip the proofs completely. Too many lengthy proofs and I have problems to get the overall picture. Then I skim over the proofs, looking for the main ideas of them. This step would be really annoying if there are too many steps. I only look at the proofs in detail if they are interesting to support my own research, I review the paper or something is suspicious¹. In this final step, I am happy about every detail that saves me time and effort.
How to address the different needs?
My usual approach for publishing proofs is as follows.
I use pen and paper to construct the proof. This results in a huge pile of unreadable garbage, but at some point, I am confident enough that my proof works.
With the ideas still fresh in mind, I write down the complete proof in a publishable form, i.e. in LaTeX², including every conversion that (in my opinion) is necessary for an undergraduate to directly understand every step.
With the fully proof at hand, it can now be condensed. For example, by presenting only very high-level steps in the main part of the paper (maybe only the final result and a textual description of the proof ideas) and a moderately condensed version in the appendix that leaves out all steps that seem trivial.
The full proof should be submitted as supplementary material if permitted by the journal (see below for an alternative).
Of course, the second step comes with extra effort compared to going directly from the pile of unreadable garbage to the most condensed form. However, it pays off in the long run:
By writing down every single step in a clearly readable form, a lot of errors are directly recognized. Otherwise, it wastes the time of your supervisor, your reviewers and (if the paper even gets published with the error) other researchers, not to mention the shame and effort when the error is eventually detected.
Even if you clearly understand your handwritten notes and your publication now, this won't be the case after a year (if you can even find them). So you have to waste time to redo your work.
If more detailed proofs are published as supplementary material, it will eventually save time for everyone.
Last but not least, it improves the credibility of your work. Even if only a very minor portion of your readers will actually benefit from or even read your detailed version, they trust you more if they see that you have a complete proof as supplementary material³.
What if I can not submit supplementary material?
In my field of research, only a minor part of the journals and conferences allow submitting supplementary material. An alternative is to submit supplementary material to e.g. arXiv.org. With good timing, you can even mutually cite the original paper and the supplementary material. You should not use a personal website because the probability is high that it will not be accessible for a long time.
Unfortunately, this is very difficult in a double blind review process. It would be much better if the submission of supplementary material is widely available at every journal and every conference.
¹ The "That has to be wrong!" effect. You might think that leaving out proof steps will help you as an author in this case. To the contrary, it increases the incentive to prove you wrong.
² If you dislike writing long and complex formulas with LaTeX: My wife is very happy with LyX.
³ No excuse for hiding a wrong proof by using an excessive amount of formulas. That will be detected eventually.