Where is the Asia-Europe divide and where can we cross it?

Istanbul is a massive city. If you wish to visit the European-Asian border but big cities are not your thing, you can take the train up to the Arctic Circle in the Russian Polar Ural.

_{(source: wikimapia.org)}

^{Asia-Europe border. Source: Wikimapia.}
In Northern Russia, there are trains between Vorkuta (infamous for its history as a penal labour camp) and Labytnangi, just outside Salekhard (the only city in the world located exactly on the Arctic Circle). Stalin tried to force prisoners to continue this railway to Igarka. Some 100,000 forced labourers were involved.

Note that unless you are the train driver, you cannot drive across the Arctic Circle here, at least not with an ordinary car, as there are no (all weather) roads (perhaps you can drive a snowmobile across?). I don't know how close to the actual Asia-Europe border the nearest train station is, but for what it's worth, HAFAS reports lots of small stops. The stop Polyarnyi Ural might be what you need — just 45 hours by train from St. Petersburg, or a direct 41-hour train from Moscow.

According to Wikipedia:

the modern border between Asia and Europe remains a historical and cultural construct, defined only by convention. The modern border follows the Aegean Sea, the Dardanelles-Sea of Marmara-Bosphorus, the Black Sea, along the watershed of the Greater Caucasus, the northwestern portion of the Caspian Sea and along the Ural River and Ural Mountains to the Arctic Ocean, as mapped and listed in most atlases including that of the National Geographic Society and as described in the World Factbook.

This border goes through Turkey, Russia, Azerbaijan, Georgia and Kazakhstan.

Istanbul, Turkey straddles the continental boundary, and is generally considered worth visiting (there is much to see there besides two continents).

Looking at Google Maps, either of the two big roads between Russia and Georgia will cross the continental border at some point (one along the Black Sea coast between Sochi and Sokhumi, the other through the Caucasus), as does the road along the Caspian Sea coast between Makhachkala, Russia and Baku, Azerbaijan.

Relations between Georgia and Russia are poor; border crossings may be closed (and in fact the road from Sochi goes through the disputed area of Abkhazia). According to Wikivoyage, the border between Azerbaijan and Russia can be crossed and there are buses, so this may be practical by car.

Numerous west-east roads in Russia and Kazakhstan cross the border.

Europe is a complex historical construct and it's not unusual to talk about the “Eurasian landmass” to highlight the fact that it has no obvious stable border.

Among the most commonly offered boundaries, you could travel to the Bosphorus, the Ural mountains and the Greater Caucasus. The Bosphorus is very easy to locate and visit, the rest not so much. It's difficult to pinpoint a specific division line but the Trans-Siberian railroad or Georgia would be places/ways where you go from one “continent” to the other.

There are also different definitions seriously considered and taught in schools. I was always taught that the border is at the ridge of Ural, and then it continues to the river Emba the to the Kuma-Manych depression. That makes Mont Blanc the highest peak of Europe. See also http://en.wikipedia.org/wiki/Boundaries_between_continents

The Bosphorus and Dardanelles straits are certain and Istanbul is a nice place to visit. You can see the statue of the bull (Zeus) which abducted Europa at Kad?köy (not that far from the ferry terminal).

In pointed out by Vladimir F in the north the ridge of the Ural mountain range is universally considered to divide "european Russia" and "asian Russia" (especially Sibiria).

Although Russia has colonized sibiria for centuries the Ural still is a very prominent dividing line. Everything changes there beginning at the availability of high quality petrol and ending in travel insurance coverage.

Since russians are grad builders of nice signposts at borders of all significances (city level to nation state level) the roads crossing the Urals are full of "Border Europe-Asia" Signs, Monuments, etc.

See here and here for some northern border crossings.