First tip: rhyming poetry is much more difficult to get published than free verse, or any other non-rhyming style. The reason is simple: rhyming is rarely done well. That’s partly because it’s difficult; the other part is probably because a lot of people take it for granted that rhyming is as simple as paring up ‘cat’ and ‘hat’. I don’t say this to put you off rhyming. It’s the truth, but if you can pull it off, go for it. Hopefully, this will give you a hand.
First of all, you have to decide what style of rhyming fits best with your poem. That’s the essential element of writing poetry, isn’t it? Finding the most appropriate vehicle for the ideas you’re putting into words. If you use rhyme, it becomes a large part of that.
The most common form, the form you probably think of when you hear ‘rhyme,’ is end rhyme—rhyme at the ends of lines of poetry. From there you need a rhyme scheme, a pattern. And you have to stick to it. You must be consistent. For instance, you cannot use AABA in one stanza, ABAB in the next, and ABCA in the third, unless you’re going to go back and do it all over again (in which case your poem should be long enough to go back and do it a third time). When you’re using end-rhyme, you also do not leave the last line of a stanza dangling. Whatever rhyme scheme you use, the last line of each stanza ought to be a participant in a rhyme.
A tricky part about picking the words is fitting them to a rhythm. Rhythm is a whole other subject, but when you’re picking your words, you have three choices that affect rhythm. Consistency is not necessarily necessary here; you have to use your judgment, paying attention to how it affects the rhythm. You can pick feminine rhyme (two-syllable rhyme), masculine rhyme (one-syllable rhyme), or you can choose to make a rhyme that consists of two words. In that third case, two words are generally used on all the lines participating in that rhyme, but it’s not a rule. Just watch the rhythm.
There are still yet more rhyming styles for you to choice from. We are back to the consistency issue; it is best to stick to a specific kind, depending on the poem. It all depends on each and every individual poem; there are few rules that can be applied, without exception, to every single poem. Now, you can use exact rhyme, selecting each word to have an identical ending. You could also choose between slant rhyme and eye-rhyme. Slant rhyme is a rhyme that is close, for example ‘maze’ and ‘face’. Eye-rhymes are rhymes to the eye, not the ear; they look alike instead of sounding alike, for example ‘through’ and ‘trough’.
Those are all options for end rhyme, but the choices don’t end there. Actually, they start there. You would have to choose end-rhyme first. But you could instead choose internal rhyme, which is when you rhyme a word inside a line with the word at the end of the line. You could, though, add a little something to any poem, rhyming or no, by inserting rhyming words into lines where you will, whether or not one of the words is at the end of a line.
I’ll mention, briefly, a third option. If you’re looking to do something more intricate, you can combine the two—end-rhyme, and internal rhyme.
ex. From Edgar Allen Poe's The Raven
… While I nodded, nearly napping, suddenly there came a tapping,
As of some one gently rapping, rapping at my chamber door
Ah, distinctly I remember it was in the bleak December;
And each separate dying ember wrought its ghost upon the floor …
There are at least 13 more such pairs of lines with that rhyme scheme (which, if you examine the entire piece, is even more intricate, having, also, and end-rhyme scheme which goes: abcbbb, dbebbb, fbgbbb, hbibbb—needless to say, Poe is truly exceptional in his art)
There is one more problem to tackle to create a successful rhyme. Now, above are listed merely the options you have when including rhyme in your poem. It has to be up to you, in the end, to make the right choices, based on the content of your poem. But if you make all the right choices and don’t tackle this last part, you’ll gain nothing. I’m not going to say it’s more difficult, but it is certainly as difficult to tackle as the rest.
It’s simply this: you have to make your rhymes sound natural. Rhymes should simply exist; they can’t sound manufactured, and they certainly cannot sound forced. If you have the perfect rhyme, it sounds almost like an accident… but not quite an accident, because that has to be how it was meant to be—if you pull it off right. The poem has to need that rhyme. It needs to be built into the poem, not disconnected, branching off, or interrupting. That’s why choosing the right words can be so difficult. You have to choose not only a word that rhymes, but it has to be the word you would use anyway. It has to have exactly the meaning, and beyond that, the connotation, that you need to complement every single other word, not just its partner in rhyme (no pun intended, trust me).
Hopefully this helped, and showed you options you haven't considered, or maybe didn’t even know you have. Rhyming is tough, certainly. But if you can pull it off, worth it. After all, it’s not everybody that can style the perfect rhyme. :)
Good luck.
Rhymes are my favorite type of poetry, though like you said in the beginning, it is rarely done well! 
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