I would have a hard time answering your first question because I like so many of them. My top three favorites are Psalm 91, Psalm 126 and Psalm 127. To answer your second question, I usually learn whole psalms, but occasionally I choose individual verses to start.
Hello Wv! There's this group, https://learnscripture.net/groups/learners-of-psalms/ , which janet23 just started. I thought you might be interested, 'cause I've noticed that you often (it seems to me?) learn passages from the Psalms. Anyway, just in case you're interested. :)
Joosep, yes, I am planning to memorize all of Philippians. I am really looking forward to it! I agree with you, Saintman, 1 John is absolutely my favorite book.