Roger Cicala wrote a 2011 completely-over-the-top article on lens cleaning that describes all the gear and methods that lensrentals uses to clean lenses, lens hoods, and lens cases.
Typically, you start with least contact to most-contact methods, but what you're more worried about is whether the coating of the front element has been physically damaged or whether it's just dirt. If it's just dirt, start with a blower or brush to remove dust. And then proceed to contact methods. The two typical ones I use would be Zeiss lens wipes (I buy boxes of them from Wal-Mart), and a Lenspen.
While you can use a liquid cleaner and a microfiber cloth, the main issue tends to be any grit that might be caught in the cloth (or on the lenspen head) that could scratch your front element. The Zeiss wipes, being disposable and individually packaged eliminate that issue, and are good for travel or on-a-shoot cleaning. [Also cleaning your glasses as well as your front/back elements and viewfinder windows all in one go.]
The lenspen is the next most convenient, since it uses a dry carbon compound (kind of like toner powder) and has a brush on the back end. The only issues (aside from keeping the head clean) is that you occasionally have to "recharge" the head by twisting the cap while it's on the pen (why you'll often see posts on messageboards by folks complaining their pen stopped working after a few cleanings) and you may have to use a blower/brush to remove any cleaning compound residue.
If neither a wet cleaning or a lenspen can remove those marks, the it's likely the coatings on the front element are damaged, and you might need to get it replaced. And if the situation that damaged your lens is likely to occur again you may want to consider investing in some UV filters as physical protection that you can more easily swap out when damaged.
Some folks calculate replacing the front element as cheaper than going through bunches of UV filters, or hate possible flare from using a filter, and prefer to rely on a lens hood for front-element protection; others prefer using the UV filters and simply removing them if flare becomes an issue. It's a personal preference on how much they're willing to pay for UV filters, how frequently they need to be replaced, and how much of an image quality hit they introduce.