My tips:
Gillette Sensor double blade cartidges- not too pricey, great close shave. Any closer with those crazy multi-blade monstrosities is not necessary, IMO, and just irritates the skin. Straight razors give great shaves too but it takes tons of practice and isn't worth the hassle unless it's a hobby.
Throw away your canned shaving cream and get a brush and shaving soap or cream in a tube. This is the most important thing by far, prepping your skin and using small amounts of stiff cream or lathered soap.
Work the cream or soap in your hand, a bowl, or on one part of your cheek at the jaw before spreading it with the brush in small circular or back and forth movements on your face, this lifts the beard for a closer shave.
Try to shave the same time every day, after a shower, as it softens your beard and moisturizes your skin, and your body reacts well to routine.
I have found that shaving downstrokes then across the grain gives me a great close shave without irritation- shaving upstrokes the second time irritates my skin too much. You'll get a feel for the lay of your beard quickly enough, I shave across from the ears to the mouth on my cheeks and from right to left, ear to ear, under my chin.
Finally, ice cold water after shaving. Never use alcohol based aftershaves, all they do is dry out your face. Some people use shaving balms or creams for afterwards, I find water lets your skin keep its natural oils and to stay hydrated.
Good luck!
PS: I tried Bic disposables once, it was gruesome, just a horrible hacking dull blade and a cheap nasty feel in your hand, plus insanely and pointlessly polluting. They are Satan's arse razor.