To clarify, do you mean reading it generally to learn about it (i.e. in translation), or reading them in the original languages?
If the former, it really depends on how much knowledge you have coming to it – if you're not too familiar with the classical world it's not a bad idea to read some secondary material before diving into the primary literature. For myths and legends, Stephen Fry's Mythos and the follow-up Heroes are a friendly and accessible way to start; for the history itself Mary Beard's books are a good introduction to various aspects of the topic.
If you're keen to dive into the actual literature either way, the most common starting point are Homer's epics – the Iliad and the Odyssey, of course, for which you will find plenty of different translations and adaptations (Robert Fagles' versions of both are considered by many to be the modern 'standard'). However if you're not a strong reader they can prove quite challenging – even if you are reading them in translation – as they are long-ish, and the language quite ornate in parts.
They do however form quite a foundational part of the classical literature that follows, both the Greek and the Roman – so you won't go wrong getting familiar with at least the content of those books through abridged versions or prose adaptations.
Ovid's Metamorphoses are weird, wild, fantastic, and occasionally horrific – so definitely not to be missed. And it's likely that you already know some of the stories in it.
On the general poetry side of things, Catullus is a popular option – mainly because his poems are often entertaining, or just outright rude.
I have a soft spot for Sappho, especially Anne Carson's version – the caveat is that none of Sappho's poems have survived in their entirety except one, so all we have are fragments of text that scholars and translators have had to try and make sense of. The result is that you wouldn't so much be reading Sappho as reading our version of Sappho – but honestly there is something beautiful in that, that Carson's collection really brings across.
In terms of prose (admittedly not my preferred form), Caesar is always a popular place to start – I'm not really a fan of military history myself, however, but hey, it's Caesar. Slightly more dynamic are the works of Cicero, possibly one of the most widely read classical writers – you get everything from scathing speeches targeted at traitors to philosophical works about rhetoric or the gods. It's hard to go wrong with Cicero, really – whether in translation or in Latin.
If you want a bit of scandalous biography then you'll want to read Suetonius's <em>Lives of the Twelve Caesars</em> – replete with plenty of gory detail about sex, murder and madness (all the fun stuff). But I wouldn't trust it by itself for any actual historical fact.
If you like a bit of drama, definitely get stuck into Greek tragedy, which is probably number one on my list of things to check out in classical literature – but I wouldn't intuitively recommend it to beginning readers in the first instance, only if because you're basically reading a play, and personally I don't think it's the most friendly format to get started with if you aren't really into either poetry or drama. But if you do want to try it, either the Oresteia trilogy or Oedipus Rex are a popular way to start.
There's plenty more I haven't mentioned (I've said nothing of Plato, for example, or Seneca, or the two Plinys), but this is more than enough to get going with ... !