As I didn't play any game in this series before, and I noticed that in Steam, Mass Effect 1 is still available. So I wonder if it is a good idea to start from 1.
I was in your shoes this past January. I decided to buy the Mass Effect Trilogy and just go for it (I'm currently half-way through ME3 after finishing ME1 and ME2).
I highly recommend that you play all games in order (ME1, ME2 and ME3) since your choices will carry on between games if you import the save files. Your choices in Mass Effect 1 will impact Mass Effect 2 and 3. Your choices in Mass Effect 2 will impact Mass Effect 3 and I'm guessing that these will somehow impact the upcoming Mass Effect: Andromeda.
This makes for a better overall storytelling experience as you can really feel your actions changing the universe of Mass Effect.
You should also consider buying the story DLC for all 3 games for a more complete experience. As Aventinus mentions, some of the story DLC fills important gaps in the story that can feel a bit lacking without them.
Keep in mind that you can't purchase Mass Effect 3 from Steam though. However (as Bowdzone mentioned) you can import your save files regardless of the platform you bought the games from (e.g. you can import your save files from ME2 to ME3 even if you bought ME2 on Steam and ME3 on Origin).
Both ME2 and ME3 have an optional comic-style introductory sequence (resp. Genesis and Genesis 2) that allows the player to (just barely) catch up with the previous scenario (ME1 on PS3, ME1+2 on Wii U) as well as make a limited selection of key choices from the previous games. Genesis 1 and 2 are available for all platforms but as DLC or directly on-disc depending on platform.
Yet, to enjoy the full breadth and "universality" of the game, an interested player should play all games by chaining them together. This is especially true for ME3 which stands as the dramatic conclusion to the Shepard epic, where many quests from the previous games are consequential to various major and minor events that play out during ME3. People complained about ME3's peculiar ending, without realising that the whole game is effectively the ending of the full story arc. Also, bonds you create with your party are not as impactful when you didn't travel the whole galaxy and back at their side, from saving each other's neck to having a party to growing a friendship or love affair.
That said, from a gameplay perspective, the ME1 engine, inventory, and controls are significantly different from and more dated than ME2 and ME3, and somehow much closer to KOTOR despite the obvious graphics quality bump. An unwary, casual player may be turned away by this challenge, especially if coming from the action games side of things and therefore may be best served by playing Genesis+ME2+ME3.
To answer your question directly, no, you do not NEED to play the prior games to enjoy ME3, it has a good story that can stand on its own.
With that being said however, for all the reasons stated in the other answers, you absolutely should play them in order, with all DLC. It will make for a much more enjoyable experience, and explain some of the subtle story arcs (Like Jokers story) and in-jokes (Make sure you take Shepherd to the club in each game. You're welcome)
Also, when you do play through ME1, please do yourself a favor and wait until AFTER the Citadel to decide whether you like the game or not... The Citadel portion can be very long and boring (Those elevators are killer!) and is not indicative of the how the majority of the game plays, it helps to set up story arcs and get you familiar with the RPG mechanics
Definitely play all three in order.
The world of Mass Effect is very complex and full of detail. The later parts assume that you are familiar with all the concepts, factions and locations established in the previous parts. Without having played the previous parts, you will soon feel lost in the vast lore (to be fair, there are some ingame encyclopedia articles which explain the basics of what you need to know, but that's really no substitute to having experienced it first-hand)
Also, lots of characters appear in multiple games and when you are unaware of what happened to them in the previous parts, you will have a hard time to understand who they are, what motivates them and why they matter. Also, by importing your savegames your previous interactions with them will be remembered and affect the story. A lot of the decisions you make matter a lot for these characters (in many cases even if they will be dead or alive), so when you just skip a game and assume the defaults, you miss out on a lot of depth.
While all three games have a self-contained story arc, they are all part of one, overarching plot on which you would miss out on when you don't play all three.
ME1 generally plays, looks and feels a bit different than ME2 and ME3, but it is still a great game and definitely worth playing.
Yes. Mass Effect 1, which was released before its sequels, was made so that you can play it without having played the subsequent games. It also starts the story of the Mass Effect series, so if you are interested in the story, it is a good place to start.
I also belive that Mass Effect 2 takes your choises in Mass Effect 1 in consideration, if you have a completed savefile of the game.
I started with ME2, and replayed it several times. I still consider it one of my favorite games of all time, having never played ME1.
At no point during the game did I feel "lost", only that past events (both from ME1 and those yet to be discovered by Shepard) were waiting to be learned via one means or another. Eventually I was able to piece together enough of the previous story to get a full understanding of what had happened in ME1, but at no time did I feel like that prevented me from fully enjoying the events of ME2. It felt more like an Easter egg, or some deeper lore such that in most games a player could normally skip learning about.