A Long Way To Fall
This is a not a time-sensitive mission, except that after you pull a gun on the initial character to get info, if you decide to kill him, you should then run on into the building before his friends start in on you. Even as you wait for the elevator inside, a Bad Guy ("BG") may follow you in, so have your combat shotgun (or other weapon) ready.
As soon as you can after getting off the elevator, open fire on the two guys there, then go through to the doorway on the left, watching for someone in the open doorway in the hall. After dealing with him, circle the floor twice to make sure you get all the BGs. You should not have to worry about any coming up behind you or from behind closed doors. Just be prepared to shoot as you round each corner.
After clearing the floor, back up the stairs aiming above you and clear the next two floors. When you get to the floor with the yellow marker in front of a door to your right, go to your left and clear the floor, then enter the apartment where three BGs are waiting.
There is a first aid kit in the kitchen if you need it. The boss will eventually escape up to the roof. Follow him up the stairs, aiming above you as before. When you come out on the roof, you can see the red marker above the boss, but not the boss himself. A BG or two may show up. After you dispense with him/them, just walk over to where you saw the red marker and finish off the boss.
Pretty easy, but due to the large number of BGs, you will want to start with full health and armor.
Taking In The Trash
This may be the easiest mission in the game. You drive a trash truck. All you have to do is remember to press A to open and close the back of the truck when bags with diamonds are thrown in. Then follow the route back when baddies start chasing you. Nobody even expects you to drive fast, so there's no excuse/reason for crashing.
For that, you get $9000.
Not a hard mission, but stock up on health/armor in case you attract cops.
Follow the targets until they crash in the park, then get out and eliminate the red dots. Go into the men's room looking for a yellow dot. Kill him and pick up the diamonds. If you have attracted cops, they have probably already been shooting at you; you have no choice but to shoot back, but once you have the diamonds, you can run across the park to the yellow marker where Ray is going to meet you.
Scamper up some big rocks, climb over the wall to the street, shoot whatever cops you have to and get to the yellow marker. This ends the mission and makes the cops go away.
Max out your health/armor and automatic weapon ammo. An irritating feature of this game is that if you run out of ammo for an expensive weapon, Niko just tosses the weapon and you'll have to pay big bucks to get a replacement, so don't get into that position.
This mission is fun because you get to battle a lot of bad guys with lots of cover. All the fun of paint ball, but with live ammo. The building (a museum) is large enough that you can snipe many of them from across the building.
When you finally exit the building, some thugs will be guarding some limos. Take them out and jump into a limo. Some bad guys will come shooting after you (literally) and the cops will join in, but getting away is relatively easy. The limo is a nice ride -- fast, handles well, and is probably bullet proof -- a good one to hang onto.
No Way On The Subway
As long as you turn left coming out of the alley, you should not have too much trouble following these guys. Don't waste time trying to shoot them.
After you get down on the subway, a train will take care of one of them, and back on the streets again, the other one crashes before too long.
Prepare with full health/ammo. This is pretty straightforward ducking and shooting. A grenade or two may come in handy. When you go up any stairs, keep your weapon pointed up the stairs.
If you need a health boost by the time you've cleared the 2nd floor, there is a health kit on the wall in the far corner of the kitchen. There's some health and armor packs on the roof, too.
After you finish, the cops will be coming. An easy way out is a lift on the edge of the roof where the last bunch of thugs were. Hop a fence, jack a car and go.