There's two ways to answer it: the tourist version, and the Morgoth-junkie version.
Tourist: You'll want to visit Beleriand during the Siege of Angband, between YSI 60 and 455. To narrow it down more than that, you need to decide what - and who - you want to see.
-If you're interested in the Eldar to the exclusion of mortals, try and arrive in ~YSI 63. That will let you sneak in a quick visit to Nevrast before Turgon leaves for Gondolin the following year. You can spend four years hearing Quenya spoken before Thingol bans it in 67; you should probably pay your visit to Doriath in this time period, too, since Thingol gets a lot more wary after that.
Then you have a bit of a wait. Most of the Elven cities are built by this time, so wander where you will, but keep an eye on the calender. You'll be pretty old by 102, when Finrod finishes Nargothrond, but you ought to make it. If you can manage to latch onto Turgon's group between then and 116 - by which time you'll have spent about 50 years in Beleriand - you can possibly live out your twilight years in Gondolin itself.
-If you're interested in mortal men, come in a bit later - during the Long Peace (260-455). I recommend timing your arrival to shortly before 355 - that way you can meet Beor before he dies. You'll have a lot to look forward to - Andreth the Wise will be born in 361, so you can meet her in her prime; Haleth the axe-wielding chieftan was born in 341, and in 375 she will claim leadership of one of the Houses of Men. Er, try not to be in their camp when that happens. Also, try not to be anywhere near Estolad and House Beor in 369, when Bereg gets tricked by Morgoth.
This timeframe won't let you visit the mortal realms of Dorthonion (founded 410) and Dor-lomin (416), unless you're very luck. The last major event you're likely to get a chance to see is Aredhel's return to Gondolin in 400 - and actually, if you can follow them, you might be able to sneak in yourself. If not, you can still witness the birth of Barahir, father of Beren.
As an alternative to the above, if you show up in the late 300s, you can still meet everyone except Beor himself. You can visit Dor-lomin and Dorthonion, witness the conversation of Finrod and Andreth, meet Beren (born 432) as a child, and maybe just about meet Hurin (441), Morwen (443), Huor (444) and Rian (450). But then you're risking being around for the Sudden Flame... not really recommended.
Unless, of course, you're a...
Morgoth Junkie: You want to arrive in 453. Make a lightning tour of Beleriand, but be absolutely sure to be in Barad Eithel (Fingon's fortress) for 455. You will have a front-row seat for the Battle of Sudden Flame, and receive eyewitness accounts of the death of Fingolfin. You will, admittedly, be trapped in Hithlum for the next 17 years, but, y'know, sacrifices.
Next up, join Fingon's army by 472. You can be present at the Nirnaeth Arnoediad, and then choose your future: either join up with Orodreth's army from Nargothrond, or (when he shows up) with Turgon. Either way, you get a decent chance of retreating to a hidden city - which you then get to watch be destroyed! Well, you can definitely survive to the fall of Nargothrond in 495, and perhaps move on to Doriath to see the Battle of the Thousand Caves in 503 (and even the Second Kinslaying in 506!); it'll be a little harder if you choose the Gondolin route, because you'll have to live until 510, a full 57 years after your arrival. If you do go that way, make sure to befriend Idril, so she can show you the secret way out!
Regrettably, there is no way for you to live long enough to witness the Third Kinslaying at the Havens of Sirion, or the War of Wrath. Sorry about that. ^_^
hS