Civilization has a very steep learning curve but if you like strategic games with a lot of depth you really should get through that and you find a really amazing game behind it. It has tons of depth and almost an endless amount of options to finish the game.
Civilization is a turn based game. Every turn you can move your units. Cities will generate production and from that they will construct buildings and units. When you start a new game you usually end up with a single settler (at least on default settings). You need to find a suitable spot to settle your first city with this settler.
Most of the time your settler already starts on a sweet spot since start locations for you and the AI heave increased bonus resources. I would not worry that much about your start location now but the best to have near are food like corn and wheat to grow your city quickly. Also gold is nice, it adds happiness and commerce once being mined.
When you build a city with this settler you get the options what to build there. You usually want to start with a worker to improve your land. A extra warrior is also nice to explore a bit around you. If you enter your city by double clicking you can set the workers on the map area to work specific tiles. Now your city is only "1" big and you can only set a single worker. In this window you can also change production and see what has already been build.
Hammers stand for production, gold stand for income and beakers stand for research. The more hammers you produce the faster things get build (except for settlers and workers which also speeds up by food). Likewise, the more beakers the quicker research goes.
To improve your land you should build workers and let them build mines(mining) on hills and farms(irrigation)/cottages(pottery) on grassland and plains. Later on you get much more options to improve land due to research. You should improve the bonus tiles first. Improving a grassland corn tile yields 5 food (along a river you get an extra commerce coin).
Once your first turn passes you get to pick a technology to research. You start with 2 of the first 5 technologies. Depending on what bonus tiles you have you should start picking the ones to improve those as soon your first worker pops out. When you hover over a research it tells you more about it. For instance animal husbandry reveals horses on the map as iron working will reveal iron and those resources might pop near you. It also opens up more building, units and much more. For instance, once you researched writing you can make a open boarders pact with the AI.
Back to the city screen, here you'll also see the happiness and health of your city. The amount of unhappiness should never exceed the happiness or your town is going to revolt. Buildings like temples produce happiness as well as lots of resources hooked up to the city. For health it's the same, pollution should not exceed the total health or you will lose food equal to the amount it exceeds. Later in the game you wont be having that much problem with the happiness but you will find cities stagnate due to pollution.
It has already been said here that civfanatics.com is a really good place for learning the game. There are some really good players on there that offer great help. You can also join the "nobles club", here a interested map gets picked and distributed to the participants and you can post updates on how you progress through that map. You can select your own difficulty but difficulties above noble should be edited with the world editor since the AI "cheats" there by starting with more units and techs.
Once you get the hang of it you should go to hof.civfanatics.net where you can participate in the all time hall of fame.
Good luck with this awesome game!