Some mechanical advice sought.

or if the rear mech hanger is welded as part of the frame (not bolted on - just replace) you can use an adjustable spanner (remove rear mech)and very gently from behind the frame, align the hanger back. But the frame should be clamped in a work stand to do this.

A mechanic once advised me that if the chain and cogs are unworn then it's always worth checking the derailleur hanger alignment; you'd be surprised how many hangers are out of alignment even when the frames are new. It's very hard to do this accurately by eye. Unfortunately the alignment guides aren't that cheap but Park do a great one and it acts as both a guide and tool to carefully bend the hanger back into alignment. It's worked for me when I've had similar problems to the one you've described.

Best way to deal with the rear mech/gear set up is to leave the chain on the smallest rear cog (leave the gear cable alone). While you rotate the chainset push the rear mech with your thumb up the cassette to first gear (using a little force initially trying to push the chain into the spokes!). While you do this look closely how the limiting screw stop the mechs movement (into the spokes) and adjust the screw stopping this with a quarter turn as required, let go allowing the chain to roll back down towards the frame. If the chain moves off the smallest cog onto the frame adjust the other limiting screw with a quarter turns. Once you’re happy (both directions) push the chain up the cassette with your thumb and let go repeating 2-5 times, the chain should never move off the cassette in either direction and be noise free.
Rear mech gear cable should be stretched after 600 miles. Stand behind the back wheel see if the rear mech jockey wheels are aligned ok, the mech is not bent (should be ok on a new bike). Loosen the cable from the rear mech (don't fully remove) turn the barrel adjust clockwise all the way in followed by 1 full 360 anti-clockwise to allow for some minor adjustment. Check the gear cable is seated correctly in the brake lever and is ‘clicked’ into 10th gear, chain remains on the smallest cassette sprocket. Pull the cable through and tighten to rear mech. All you now need to touch is the rear mech adjustment barrel (while you rotate the cranks) nothing else!! 180 degree turns on the barrel until the chain moves from 10th up to 9th sprocket followed by small turns until the chain rubs against the 8th sprocket you’ll be able to gauge this from the chain noise. Once this is initially set you can then use your gear level changing gear up and down the cassette making minimal adjustments (clockwise/anti-clockwise) through the barrel adjuster so the chain moves smoothly and quietly between each gear change. Some small barrel adjustment will be required but remember all the barrel adjuster is doing is altering gear cable tension and the adjustment is always done with the chain on the 10th, 9th, 8th cassette sprocket area. Google ‘Shimano tech docs’ for picture advice. Don't forget to clean then oil your chain.

Being a few months old and if all else fails. Give the bike a good wash and take it back to the shop for it's 'free' (if it hasn't had one)after sales service. Even if it's had a service any descent bike shop should further assist you, see you right without further cost to you. Customer service is what this is all about. While it's there the mechanic can show you how easy it is to maintain Shimano gearing which once settled in and looked after should never be a bother. If you've bought the bike online and having limited maintance knowledge the initial 'bedding in' period will cause you problems, we've all been there. Always buy bikes from a quality high street dealer to help you with maintance and warranty issues.

http://youtu.be/SkzvfCaIbyQ

Hopefully that video will help you out. Thats what i used when i was stuck on adjusting someone's gears after they buggered it all up and had tried to adjust the shifters without knowing that they were playing at

I like to think I know my spannering, but there are times when it is all going a bit FUBARd I put everyhing back to factory settings, go and make a cup of tea then start all over again. Often the only way if it aint right and you are positive nothing is worn .