There are many poetry forms for writing short poems. I have touched on the Japanese forms of haiku and tanka, separately, because they are so popular.
A couple of things to keep in mind ... While there is still an interest in rhyming metered poetry (The Society of Classical Poets, pretty much, publishes nothing but – and there are a few others that publish a smattering) free verse is considered to be far more contemporary. Indeed, the only instances where I have noticed short poems rhyming (as opposed to longer poems, which I have seen a bit more of) has been in the case of satirical or funny poems. (In which case, a good deal of the humour would have been lost, if the lines didn't rhyme.) I enjoy a rhyme – done well – myself, but that's the way it is. However, what I have noticed, is that a number of poems are employing what I have come to think of as a 'slip-and-slide' rhyme. That is: a rhyme/near-rhyme cascading through a poem ... not at the end of lines, but rather ... at the end of a line ... then the third or fourth word in .. and so on. It does tend to impart more of a lyrical vibe, without leaving a 'dated-feeling' in its wake. I mention this because, if you are writing for yourself and your family “rhyme away”. If you are hoping to get published, it's going to be a tougher go-of-it, with rhyming poetry.
Also, it is a good idea to check out on-line examples of previously published short poems.
There are several journals that publish these forms, I would say that most of them stick to the Japanese forms but, Shot Glass Journal is an exception (and I'm sure there must be others). Check it out at:
Short Glass Journal: an online journal of short poetry:
Short Glass also has a wonderful glossary and descriptions of all the short poetry forms, it has used on its site. If you are interested in trying to get a piece published, I have found them to be very supportive.
Though having left web publication, in favour of print, hedgerow: a journal of small poems still posts a sampling of fine haiku at:
A print journal, currently on a long hiatus Skylark Tanka Journal, nevertheless showcases a small – but wonderful – gallery of tanka and tanka art at:
Sadly, a hundred gourds is also no longer publishing but, happily, years of journals can be assessed by hitting their archives tab, where you will find splendiferous pieces of haiku, tanka, haiga, haibun and renku (colaborated works) at:
And there is lots to look at on NeverEnding Story – the first English-Chinese Bilingual Haiku and Tanka Blog at:
I'm sure that I've just scratched the surface, here.
Happy Poems to You.