I'm a veteran of this game, so my strategies may be a little ambitious for a starting gamer. If you need some advice drop me a PM.
What I usually do as the English is take York, then Dublin and then Caernarvon with one army, and Rennes, Aquitaine (its to the south, on the French west coast) and then finally start working my way through the rebel provinces on the borders between France, French England and Germany. If you're lightning quick you can seize everything south of Scotland, a massive chunk of France, and Alsace Lorraine, as well as part of Switzerland, without drawing the ire of the big factions. While doing this, I negotiate perfect relations with France, ensuring they are on my side and go after me last.
However, if you're a newbie to the game, I doubt you'll be able to do that; you have to be prepared to fight rebel armies which significantly outnumber your side and you need luck. What I'd suggest is taking York, Caernarvon and Dublin, before striking Scotland and destroying it quickly. Rather than try to take on the massive and dangerous France, try advancing up the coast while maintaining relations that are as good as possible with both France and Germany; get mutual military access and perfect relations with France if possible (it's doable.) Once you've taken everything you can (try to at least take the low countries) I then invade Norway and Sweden before taking Denmark. Once you've got Scandinavia, northern France and the British Isles firmly in your grasp, you'll have the wealth and military might to go breaking heads anywhere you like; very few factions will have the strength to resist you by that point, as long as you keep a good ratio of castles to cities. Generally I have one castle per region of the empire; in this case, one for France, one for Britain and one for Scandinavia. This lets me churn out armies at a reasonable rate, while my cities fund those armies.
Gates? York is a village.
In that battle I tend to use my peasant archers to pick off the spearmen (run away when they sally) before sending in my knights to hack down the peasants. If you make sure your knights wade right into the middle of the fighting by clicking forward to move them into the midst of the melee (DO NOT DO THIS AGAINST SPEARS) then the knights cause about three times as many casualties as wen you just leave them dancing around the side of the formation; cavalry have an odd habit of just standing on the edge of melee getting swung at after the initial cut-down... and get stabbed at and cut down themselves. If you move them so they're embroiled in the enemy unit then they start using their powerful strikes which will butcher peasants and other light infantry. (Be wary against heavier infantry; you really need a charge cycle for that.)
Ouch... the running ahead of your reinforcements problem. While you can win that battle and thus get the city a turn early, it's usually best to wait for the reinforcements. What I usually do is get the reinforcements, take Rennes, and then dash down south to Bordeaux castle and take it, before using the resulting mountain of income to build an army to start taking Bruges.
Use your archers more during sieges; its entirely possible to win Rennes in an assault with no casualties outside your Generals bodyguard by using your archers to kill most of their army. (Ok admittedly when I tried this, my general was the only man left standing in his bodyguard... but he killed those bloody mercenaries!)
I fully admit I tend to use my bodyguards as free, and very deadly, fodder. There's nothing like obliterating an entire unit of spear militia with one charge and losing three men who will replace themselves for free in as many turns.
Also, if I may offer a bit of advice: remove skirmish from your archer units and simply pull them back manually when the enemy gets close. It's a bit of a fiddly thing to do, as they will always play their firing animation if they started it, so telling them to haul ass just before they're done shooting one volley is the most effective and they fire most of the shots anyway.
I would as an aside advise you to upgrade Nottingham to the highest possible archery range and get longbowmen, pronto. They outrange everything you'll be facing for a long time (until you hit Italy or the Moors, and by then you should have numbers on your side) and can slice up non-heavy units with ease.