Rules for entering Europe depend on where you are going. Citizens of EU countries and the European Free Trade Association (EFTA) countries (Iceland, Liechtenstein, Norway and Switzerland) can travel freely throughout the continent (except Russia, Belarus and the Caucasus), so the following applies only to non-EU/EFTA citizens.
If you are entering a Schengen country and you plan to visit only other Schengen countries, you need only one Schengen visa.
The 90 days visa-free stay applies for the whole Schengen area, i.e. it is not 90 days per country as some assume. Citizens of the above countries who wish to travel around Europe for longer than 90 days must apply for a residency permit. This can be done in any Schengen country, but Germany or Italy are recommended, because many other countries require applicants to apply from their home countries.
Non-Schengen countries, on the other hand, maintain their own immigration policies. Consult the country article in question for details. If you wish to visit a non-Schengen country and return to the Schengen area, you will need a multiple-entry visa. Cyprus, Ireland, and the United Kingdom are EU members, but they are not part of the Schengen Area while EU members Bulgaria and Romania are currently in the process of joining the Schengen Area. To add confusion Switzerland, Liechtenstein, Iceland and Norway are not EU members but part of the Schengen area.
The implications of this are simple: countries in the EU maintain similar customs controls. Therefore, you do not need to pass through customs when travelling to a non-Schengen EU country, but you may need to pass through immigration controls. The converse is true for non-EU Schengen countries: you must pass through customs, but not immigration.
The largest air travel hubs in Europe are, in order, London (LON: LCY, LHR, LGW, STN, LTN, SEN), Frankfurt (FRA, HHN), Paris (CDG, ORY), Madrid (MAD), and Amsterdam (AMS) which in turn have connections to practically everywhere in Europe. However, nearly every European capital and many other major cities have direct long-distance flights to at least some destinations. Other, smaller airports can make sense for specific connections: for example, Vienna (VIE) has a very good network of flights to the Middle East and Eastern Europe, while Helsinki (HEL) is the geographically closest place to transfer if coming in from East Asia.
The Trans-Siberian Railway from Beijing and Vladivostok to Moscow is a classic rail journey. The Historic Silk Road is becoming increasingly popular with adventurers trying to beat down a new path after the finalized construction of a railway link between Kazakhstan and China. This new Almaty–Urumqi service runs twice per week, and Almaty is easily reached from Moscow by train. Other options include several connections to the Middle East, offered by Turkish Railways (TCDD). There are weekly services from Istanbul via Ankara to Tehran in Iran, but the services to Syria and Iraq have been suspended, hopefully temporarily, due to the troubles in those countries.
It is still possible, but expensive, to do the classic transatlantic voyage between the United Kingdom and the United States. The easiest option is by the historic, and only remaining Ocean Liner operator, Cunard Line, but expect to pay USD1,000-2,000 for the cheapest tickets on the 6 day voyage between Southampton and New York that sails around 10 times per year in each direction. If your pockets are not deep enough for this price range, your only other options of crossing the North Atlantic are pretty much limited to freighter travel and "hitchhiking" with a private boat.
Most major cruise ships that ply the waters of Europe during summer (June - September) also do cruises in Latin America and Southeast Asia for the rest of the year. That means those ships have a transatlantic journey twice per year, at low prices when you consider the length of the trip (at least a week). These are often called positioning cruises. MSC has several ships from the Caribbean to Europe at April and May.
There are several lines crossing the Mediterranean, the main ports of call in North Africa is Tangier in Morocco and Tunis in Tunisia (See Ferries in the Mediterranean for more details), but there is also a little known option of going via Cyprus where you can use Louis Cruises crossings to Port Said in Egypt and Haifa in Israel as a regular ferry service. Keep in mind though, that you can only do this on routes out of Cyprus, and it requires special arrangements – Varianos Travel in Nicosia seem to be the only tour agency offering this option. If you're time rich, but otherwise poor, it may be possible to "hitchhike" a private boat.
You are legally allowed to bring through the EU border limited amounts of tobacco (exact numbers depend on your arrival country) and 1 liter of spirits (above 22% alcohol) or 2 liters of alcohol (e.g. sparkling wine below 22% alcohol) and 4 liters of non-sparkling wine and 16 liters of beer. If you are below 17 years old it's half of these amounts or nothing at all.