Velobase gives the dimensions of bars & stems. Usually, the safest bet is to have bars and stem from same manufacturer once you're sure the stem has the right sized clamp for the bars, but any 26mm stem will work with any 26mm bars. Checking on Velobase you can see what works with what.
As for stem lengths, it depends on your position and how fussy you are about fit. My guess is a 90mm or 100mm should work fine for someone your height on a 51cm frame (I'm similar size and ride similar size bikes and use 90mm). If you're the sort of person who fusses around with a plumb line to get the knee over the pedal spindle right, it may take some experimentation with stems to get the reach right. If you don't know what I'm talking about, trust me, ignorance is bliss. Just set up the saddle however feels comfortable and try a 90mm or 100mm stem for starters. You'll inevitably tweak the position little by little over time.
I personally prefer 42cm bars (c-c), but the bar width is more a personal choice. Beware that the Italian manufacturers don't all measure their bars c-c (see Velobase again once you've decided what bars you like). Bar shape is also very personal. Some peeps like deep drops, others prefer shallow. Basically, you can't go too far wrong with a traditional shape like a Cinelli Giro d'Italia or 3TTT Tour de France bend, but you may decide you want a deeper drop than that classic shape. If you're using brifters, you're better off with modern compact bars that give a more comfortable position from tops to shifters.
Don't worry if you don't get it all 100% perfect first time around. Nobody does.
Btw, I've some lovely 3TTT stems and bars available. The stems are Record 80mm, 85mm, and 90mm; the bars are two 3TTT TdF, one's 40cm and single grooved, the other 42cm and double grooved, and a Deda 215 42cm and double grooved. They're all interchangeable. PM if interested.