I’m a beer lover, who is trying to venture into drinking other alcoholic drinks. I can’t get myself into liking wine. Mixed drinks, I think I can do.
For reasons I’d rather not go into, I’m not looking for drinks that a 21 year old club whore would drink. I’m looking for drinks that have a fun yet more mature and even a slightly sophisticated image associated.
By the way…is a mixed drink and a cocktail the same thing?
Yes.
Starts with the basics - i.e., standards that are no more than 2 - 3 ingredients:
Branch out from there. Of those mentioned a Rum and Coke is the sweetest and easiest to start with, without looking like you having a foo-foo drink in your hands…
sweet vermouth on the rocks with a twist
gin and tonic for sure. vodka/grapefruit juice (greyhound) is another personal favorite. soda mixed drinks are usually too sweet for my tastes. on-the-rocks and fruit juices are the best way to go.
Old fashioned
Gimlet
Manhattan
Martini with gin and no fuckin’ vermouth (and BTW, not all gin is created equal: see older threads for the gin wars)
If you like savoury, try a bloody caesar. They are dee-licious.
The great thing about a gin and tonic is that it’s not just delicious - it’s the Official Drink of Imperialism. 
(Seriously - the quinine in tonic water helps ward off malaria, though there’s not enough in the stuff you buy at the supermarket to help all that much.) Fix yourself a good gin and tonic, sing God Save the Queen, and go conquer something - it’ll be a rich, full evening. 
Gin and tonic was the drink I started off with, too. I don’t really drink much anymore but when I did, it was always a good stand by to order at a bar.
ETA: Loved making those quinine jokes, too, Mr. E.
Fill an on-the-rocks glass 3/4 full with ice
Fill glass about 1/4 full of good vodka
Drink
Repeat as needed
Greyhound or a good Bloody Mary.
White Russians, dude.
I don’t know about it having a “mature and sophisticated image associated with it” (are you getting into this because you want something enjoyable to drink or because you want people to notice what you’re drinking and think you’re really mature and sophisticated?), but my favorite mixed drink is the Tokyo Tea. It’s a Long Island with Midori instead of Coke. If made right it’s very strong but also quite palatable.
A good Rye Manhattan. Which means Russell’s Reserve Rye at the very least. No Old Overholt.
It always makes me think of Rome…
Gin and Tonic, White Russian, Bloody Mary, Ginger and Whiskey (get a double shot of whiskey) I think are all good choices for adult mixed drinks. I would avoid starting off with drinks that don’t contain a mixer, you likely will not end up liking them (although eventually you probably will, you got to start off slow though).
Whiskey sour.
Whiskey (any kind) and Coke or 7 Up.
I personally would warn you that a White Russian is not a drink, it’s a dessert. It’s boozy coffee milk, you can’t be drinking that all night. Bloody Marys/Caesars, while also delightful, are also very heavy, and more of a brunch thing, in my mind.
I’d start with WordMan’s suggestions, in my mind you can’t beat gin or vodka and tonic for a summer drink, anyway.
Don’t horse around. Go straight to the classic, the Martini. Gin or vodka is acceptable, but it **must **have a tiny amount of vermouth. Otherwise, it’s not a Martini; it’s just a big shot.
Vermouth is for girls.
When I was of an age to care about such things, these particular few would have been seen as drinks appropriate for an older generation, like a Rob Roy or a Rusty Nail. Have they come back into fashion?
Diamonds02, one word of advice: I’m admittedly a lightweight, but the times when I’ve gotten sickest have been when I switched up drinks over the course of an evening. My suggestion is that you try no more than one or two things, in the course of an evening, and then possibly switching back to beer if you find you don’t like them. Don’t keep trying four or five different liquors in one evening.
Finally, some drinks are traditionally associated with certain times of day. Brunch suggestions would typically be a mimosa, a kir royale, or a bloody mary. After-dinner drinks are less likely to be mixed (except possibly with coffee) and would include brandy, cognac, or a flavored liqueur like kahlua.