England's issues weren't hard to see.
We're vulnerable down the flanks. On the right this is somewhat unavoidable with the squad we have; Glen Johnson is a fullback who can't defend (and can't really attack either) which means we're always open on that side. But on the left it was a tactics/selection issue. Rooney isn't a winger and likes to come centrally which meant that Baines was left completely exposed and normally stuck in a 1 vs 2 whenever Italy could come forward at pace.
Secondly, we had a virtually non-existent midfield. It was frankly embarrassing the way the Italians could slice right through it whenever they decided they were tired of passing it around the back. Offensively it was little better... until the last 15 minutes when Italy dropped back and decided to defend there was nothing connecting our midfield and attack which meant it was very easy for Italy to snuff out our moves. Henderson is a useful workhorse but lacks quality while Gerrard is basically past it and the fact he's better at the self-described “quarterback” role then people expected just means that he's better than people expected, not that he's actually good.
Both Italian goals came from those above issues.
The first was a result of our midfield not closing down quickly enough and focusing too much on Pirlo. The second because Baines was outnumbered on the left and Johnson missing on the right when he should have come in to block the header.
Offensively, when out front four did get the ball there was generally a lack of precision outside of the goal itself. Rooney delivered one beautiful pass but otherwise huffed, puffed, shot wide and otherwise did nothing. Sturridge likewise. Welbeck likewise although his defensive . Sterling, who was our best player in the first half clearly ran out of steam in the second. Barkley didn't add the drive we expected but to be fair he was largely marked out of the game, Whilshire did his usual thing of dribbling into a cul-de-sac before losing the ball and Lallana showed flashes but didn't really have enough time to make an impact. Lot's of nice looking direct runs but in general a lack of end product with too many poor decision, poor shots and poor crosses.
As for Italy... in honest there was nothing special there. Ballotelli managed one simple header after being left unmarked through incompetence rather than excellent movement, Pirlo managed a couple of moments of magic but was far from his conductorial best and on the whole they succeeded because of our flaws as opposed to their own brilliance. They exploited our flaws ruthlessly and defended fairly well but I don't think any of the major contenders will be particularly scared; Pirlo firing long balls at Balloteli as he tries to break the offside trap is their main tactic.
So what do England need to change for the next game? Don't use Rooney on the wing. He's even worse there than he is at 10... and I don't think he's a great 10. Honestly, if you're going to play him I think you have to play him as the out-and-out striker... and if you want to play Sturridge there that means no Rooney. Use Welbeck on the right, Sterling or Lallana at 10 and if you play Sterling there then either Milner (somewhat negative but he protects Johnson) or AOC (assuming he's recovered from injury) on the right. If Lallana's at 10 then Sterling should likely get the nod on the right. Barkley remains as the super-sub option to inject pace and directness late on.
As for the midfield, we're not exactly spoilt for choice. With Gerrard being captain he's essentially undroppable which means Henderson likely has to get the nod to make up for Gerrard's lack of movement or workrate. Whilshire isn't defensively solid enough and has arguably regressed from where he was as an 18 year old while teaming Lampard and Gerrard when they were at their peak didn't work... it certainly won't now. I suppose you could use AOC as a more direct option but I think he'd be wasted having to sit back quite so much.
Oh... and if we're chasing a game, bring on Lambert. He offers something different as well as brilliant hold-up play, passing and a great understanding with Lallana.