Derakon beat me to it... but I'll still post it for the kicks.
Yeah, Civ 4 changed quite a few mechanics
1. I believe F2 is the financial adviser. You get a number of units at no cost depending of the difficulty level. Then you also get a number of free units depending on your total population points. Everything that goes over that value costs money. 1 gold per unit, but the cost is further reduced by the difficulty level. The vassalage civic gives you a few extra free units (but costs more civic upkeep). The pacifism religious civic also incurs an extra cost for you military units (population and difficulty levels again influence those values).
Most of these values still stand in BTS
http://forums.civfanatics.com/showthread.php?t=158130
2. The happiness slider is now the culture slider. I believe you need to research drama to unlock that. You get 1 happy face in every city for every 10% (of you total commerce). It also converts that 10% into culture for each city. Some buildings like Theaters and Colosseums increase the number of happy faces the slider provides. 20% culture gives you 2 faces. If you have the two buildings I mentioned in a city you will get 2 + 2 + 1 happy faces.
3. Corruption is gone! No shields, ahem, hammers are wasted. Maintenance costs you gold. Buildings do not increase the maintenance of a city, only its distance from a palace, and the total number of cities... there are also corporations and colonies / vassals that might add to your maintenance.
4. Open borders allows you and the person you sign it with free access to each others territory. You can send missionaries to spread a religion and you can settle after his land. In Civ 4 if you enter a Civs territory without having Open Borders you declare war. Coastlines can also be blocked to your galleys unless you have open borders.
Open borders also generates commerce for both of you since cities now have trade routes. A trade route with another civ's city yields more that a domestic one (coastal cities generate a lot more commerce through trade routes)
5. To kill a longbowman on a hill city you will need strong city raiders and preferably siege units. The longbowman will have a defense of 6 + 25% city + 25% hill + 25% fortification + 0-100% city culture / walls / castle. Add to all that city defender promotions and you get a really, really high number.
Bring trebuchets or catapults and use them to bombard away the city defense. Then send the siege units in first to damage the defenders (they damage multiple units). Since siege units cannot kill units in BTS you will also need an escort to finish the job. I recommend Macemen, Knights or Swordsmen (if you poor) for the job. Give them city raider promotion if possible and always check the combat odds.
6. Each forest in the fat cross of a city gives 0.5 health. So if you have an odd number feel free to chop. Flood plains lower health by 0.4 and jungles by 0.25 for each tile. Always round down.