1 - Play the map. There are situations to play peacefully (room for 6+ cities without war), and situations in which one should play bloody. Learning to do both will help you immensely moving from level to level. The diplomacy skills of a peaceful game are of marginal help when playing for domination/conquest at low levels, but are an essential part of any strategy at higher levels.
2 - Don't be a wonder whore. It's easy to do at low levels, but it will not improve your overall gameplay to build every single wonder. There is no one wonder you need to win (on most maps) and building things like the 'mids in every game can sometimes convince a new player otherwise.
3 - Build more workers than you think you need, then build some more. 1.5 per city is a good ratio, with more needed if you have lots of jungle and marginally less if there is very little forest/jungle.
4 - (related to 3) Avoid working unimproved tiles.
As to civs to try: I suggest trying a variety of civs/leaders. The easiest is Huyana Capac (inca), who can build his UU on turn 1, and if all you do is build a bunch of quechas and go kill the two nearest civs and take their capitals (very easy to do), you will have a hard time losing up to prince level. Some better ones with which to learn a variety of skills: Mehmed (ottoman), Darius (Persia), Willem VonOrange (Dutch).
Really though, it is good to play with a variety of traits and learn which ones favor your style of gameplay. Different people play the game differently, and certain leaders work better for crowding (placing many cities close together), REXing (settling as many cities as possible as early as possible), rushing (building a mass of early military units and killing 1-2 neighbors), wonder whoring (building as many wonders as possible), Cottage economy*(building many cottages to fuel

), Specialist economy* (building farms to work specialists rather than relying on cottage

) and just about every other imaginable way of playing.
* Not recommended as an exclusionary tactic - hence a cottage economy should use specialists and a specialist economy should have some cottage cities.