This is a difficult question to answer succinctly and accurately, because both rhythm and poetic meter are extremely varied. Even within a purely in top-40 pop context, the way Jay-Z sets lyrics to music is very different from the way Sia sets lyrics to music. Music is often characterized by a lack of "hard and fast rules", but I think that is exceptionally true in this case; it is much easier to get audiences to accept unusual syncopation and syllable misemphasis than it is to get them to accept unusual tonalities.
Before you read any further, it is probably best you read this answer to How can a non-musician recognize anacrusis?, which demonstrates anacrusis by explaining the concept's relationship to lyrical stresses. One really good exercise would be to literally count out the one-two-threes of Happy Birthday as-in that answer, and actually sing the first syllable of the song on the first beat of your count, with full emphasis; the example of how lyrics should not sound might be more helpful to you than the example of how lyrics should sound. It is important you take note of the idea that a bar is a series of beats, and each bar typically has the same number of beats, in the same order if the beats vary in length -- and even when this is in some way not true, there is usually a recognizable pattern. There is no obligation for a song to be in fours, and if you choose a traditional poetic meter, you may want to experiment with an unusual number of beats to a bar to give the lyrics a faster, choppier and/or more mechanical feel, if that effect is appropriate.
I would suggest that you write your lyrics first, and then your music -- whether or not you should constrain yourself to any particular poetic meter is up to you. Traditional poetic meter sets tone because it comes with other genre conventions, and it also provides an interesting artificial framework for a skilled poet to fill with naturally-flowing language -- whether this appeals to you is something only you can say (disclaimer: I think the Writing Stack Exchange crowd were somewhat wrong or oversimplificatory about some of the music stuff; no doubt they would say the same of my views on poetry).
Similarly to the anacrusis exercise, speak your lyrics aloud and note where the natural syllable stresses lie. These stresses are -- unless you consciously decide otherwise for artistic effect when you feel more confident -- your beats. Say your lyrics aloud again, except this time try to keep those beats at a constant-ish rate, or whatever feels "rhythmically natural" -- I'm oversimplifying and handwaving tremendously here, asking you to appeal on some level to instinct, but at the end of this process, you should find you have two classes of stress, stronger and weaker, and the strongest stresses represent the first beat of your bars, which is usually (oversimplifying again) where things like chord changes and "new sounds" go, and when the bass drum kicks in. Melody notes here will usually not be overly dissonant, and/or will often resolve quickly.
Also, what you first come up with may only be a skeleton rhythmic cadence -- unless your original meter was quite varied, or you want the lyrics to come out mechanically, there is a good chance you might want to "pull this rhythm around", deliberately move notes forwards or backwards a bit in time. The first minute of John Coltrane's Chim Chim Cheree is a good example of the sort of rhythmic variation you might want to introduce. I often use my phone to record several spoken-word recitals of the same lyrics in very different rhythms before choosing one.
Also, it's okay if some songs a "more lyrical" and other songs are "more musical", or don't even have lyrics, too. I love The Killers, but I could never defend their lyrics, they just performed their pop-rock really well with a really cool sound.