Choosing a Period
You can wargame any period in the past, future, or fantasy! If you're into fantasy or science fiction, then I suggest gaming in the 'world' that captures your imagination;
There are published or free rules for most well known genres. Below I provide a guide to the differences in wargaming historical periods. Ancient and Medieval warfare tends to be less mobile than later battles, the way you set your forces up and the terrain (model scenery) you have is critical, after that armies tend to advance and the best (or most lucky!) units win the combats. It can be a bit repetitive after a while, so there are different scenarios to play to give you a challenge instead of a straight battle. However, this is a very colourful era, with great variety in appearance and style of play between different armies, with some exotic troops such as scythed chariots, elephants, camels and war wagons.
Renaissance, Marlburian and Napoleonic Warfare is a 'scissors, paper, stone' contest between infantry (armed either with proportions of pike and musket or all with musket and bayonet), cavalry and artillery, with different ones dominant during the different periods. The art of warfare is to force the infantry into a disadvantageous position or formation, and then take advantage of this. Armies by now have uniforms, still in some cases extremely colourful but mostly similar. Naval Warfare in this period is different and challenging because until modern warfare ships moved forwards and fired sideways. This makes for a unique game, as in most games units attack in the direction they move, and requires careful planning. The ships are sail powered, and rely on the wind direction to move as well!
Warfare accelerated with the introduction of tanks, and moved into three dimensions with the addition of air forces. This is the period for gamers who like a wide variety of capabilities in different units. By this time units are usually camouflaged (less colourful, but in most cases easier to paint), and some may be very powerful or invulnerable to others.
Choosing a Scale or Size
There is a bewildering array of different sized models available; Most commonly accepted scales range from 6mm to 28mm. These must be the same size as the armies you will play against! Also beware that different rules may require them to be based with a certain number of figures on a specific sized base, often as long as both armies are based in similar basis most rules sets allow for some flexibility.
If you have a choice, 6mm gives the best 'army' look with lots of figures, but the individual painting on models will be quite simple. 15mm is a compromise, and 28mm allows exquisite sculpting and painting of individual models, but at the expense of the number on the table. Historically 20mm was a very common scale due to the availability of cheaper plastic figures and indeed vehicle and plane models but is less popular now with the exception to WWII.
Before choosing an army the best advice I can give is to find and join a club, even if this means you have to travel a long way and can only make it once a month. There you can try out lots of games and find what you enjoy, and work from there. You can play big battles with small numbers of figures, small battles with big numbers of figures, and there are lots of different rules with a trade off between playability and accuracy.
Choosing an Army
Read up a little on the period that you will play in, and select an army that appeals to you! It can be helpful if lots of other players do not already have the army but is not always a major issue.
One of the great parts of wargaming is taking part in campaigns, not just winning a battle with your army but leading it through a period of time, planning where to fight your battles and then conquering a country at the end. Here is an account of one club's replay of the well known contest for the English Crown in 1066, with history being dramatically rewritten.
You will need:
Someone to play against;
Some dice. Wargamers use not only 6 sided Dice, but sometimes 4-, 8-, 10-, 12-, and 20 sided too!;
Some cardboard (cereal packets are great for a cheap start!) to represent the units;
A playing area. Size varies but 4' (120cm) square is a good start;
A set of rules. There are many to choose from!