ZvT should be fairly easy for you to win. It's definitely in your favor. I'm assuming you aren't doing something stupid like a super-stack or "LOL $$$$" map, where the Zerg player will always be at a disadvantage because the cheapness of their units is somewhat negated by the resource abundance. I'm also assuming Brood War, not the standard game.
Micromanagement isn't a huge deal for Zerg. What is big is booming. Any time you've got a few hundred minerals, you should be building another hatchery. Prioritize other buildings only till you get the hydralisk den and evolution chamber, then work on getting about 8 hatcheries (including your starting hatchery, which should be a Lair by now) in your main base. Pump hydralisks like crazy. You will probably need more resources. Go expand into your standard expansion, and maybe his if he's slow. When you have the chance in there, start shifting over to flyer production--not mutalisks unless you know how to use them, but definitely guardians and scourges (maybe a few devourers if he's using Wraiths, as their cloak is useless once you get acid spores on them). Ignore stuff like ultralisks, as they cost more than they're worth. Queens can be useful in
small numbers--broodling a Siege Tank within a mass of them and they'll usually friendly-fire one or two other tanks. Defilers can be useful in small numbers; Plague, not Dark Swarm, is their most useful ability against Terran, as repairing mechs can be expensive. Plague also does a hell of a nice bit of damage to Battlecruisers if he's one of those types, and a Scourge bomb can usually finish the job and make him waste a lot of minerals and gas.
I'm not sure why Science Vessels are a problem for you; Irradiate does a lot of damage, but you should be able to Scourge any SVs that get too close.
The key here is to throw massive numbers at him until he chokes. Don't bother with overlord drops or other attempted tactics; they're not that important. Outmass him and keep hitting him. Don't give him time to rebuild. You should have the ability to pump out 24 hydralisks every time each hatchery hits 3 larvae--if you don't have the money,
expand more until you do. You cannot sit in a hole as
any race in Starcraft, but it's much, much worse for the Zerg not to own multiple expansions and have really, really good economy. Yes, this means you have to "control a lot of the map." That's how Zerg is played. If that's a problem for you, don't play Zerg.
What should I do when he makes goliaths + tanks and a couple of science vessels? I found I can't really handle them, late game.
Shit, there's plenty of things to do. Mech pushes are so easy to break. A scourge squad will overwhelm Goliaths enough that you can gank the science vessels. Queens can Broodling Siege Tanks and Goliaths, which (if the siege tanks have shifted) is, again, a great way to cause friendly fire, and to instakill a unit otherwise. The key, though, is to burn down the science vessels, as lurkers or other mid-level units can generally gank the mechs as long as the detectors are down. And if you're getting desperate, a Zergling flood against them should probably be okay.
Complaining about "not being fast" is not going to get you anywhere with Starcraft. It is a game where actions-per-minute
matter. It's like complaining that you can't catch a football, when you want to play a wide receiver. It's just how it's played. Anyone can learn it.