No, magnetic clouds disable abilities and plasma storms disable strike craft. A starbase in plasma storms are downright annoying to deal with. A properly equipped and supported Transcencia in a plasma storm should be impervious to attack from anything except a swarm of Ogrovs with Targeting Uplink or an Orkulus. Fortunately, as tough as an Orkulus is, you should be able to push it down through sheer brute force if you hit hard enough.
Last time I encountered an Orky at a plasma storm, I played it cool so the enemy thought I was building up, then suddenly showed up with 20 Destras, a few guardians, and a Progen. Tanked my way through the starbase and brought it down. If you hit hard and fast enough, Orkies die to frigate power.
Anyways, on to the topics raised by the original poster:
what's a good build order to start off the game with?
If you're playing without quick-start, you should always build your capital ship and your extractors right away. If you anticipate a longer game with the enemy starting further away, upgrade your homeworld's population. If you're starting with quick-start on (requires the Entrenchment expansion pack) all these things are already purchased for you.
Definitely expanding your empire is your top priority, as is scouting. Depending on how far away the nearest enemy is, you'll either way to get two civic labs (for ice/volcanic planets) or two military labs (for military units).
Stay lean on techs. Focus on high priority and low-cost technologies. Expensive technologies are only worthwhile for very large empires, and even then you don't want to get them all. Focus more on growing your empire, your fleet, and adding more infrastructure (like trade ports).
Another thing that I've been getting my butt kicked by is the pirates...early game is it better to sic them on someone else with bounty or fight it out?
In the original game and Entrenchment, it's definitely worth it to fight the pirates. You can easily fend them off with a single capital ship, a few turrets, and a repair platform. In Diplomacy, pirates are much deadlier and bidding makes more sense. Honestly, I'd recommend turning the feature off while you learn the game.
What's an ideal fleet makeup
There are lots of different approaches. As Vasari, the Assailant is your best damage-dealer, while the Sentinel is your best damage absorber. Nothing wrong with carriers laden with fighters and bombers, either. As for durability, the best way to keep your fleet alive is to pack firepower and lots of it (this is especially true for Vasari). Pound the enemy into a pulp before they can do the same to you. You can use Overseers for healing if you like, but those are best reserved for capital ships.
As for capital ship to fleet ratio, generally your fleet size research should be one step ahead of your capital ship crew research. That's a generalization; I frequently run capital ship heavy and capital ship light depending on my situation. Capital ships give you less firepower for money spent, but more useful special abilities, while frigates give you muscle. It takes some experience to balance the two.
How do you defend your planets?
Your best defense is the scout; know where and when the enemy will attack before he gets there, and then have your fleet in place to defend. The best way to support a defending fleet is repair platforms. These are very inexpensive, but can greatly extend the longevity if your units (particularly capital ships) in a fight.
Starbases are great for choke points and other critical "junction" planets, but they won't be able to defend your entire empire. The best defense is always a mobile fleet, and starbases support the fleet.
Now, the AI can be easily lured to attack starbases that are simply too strong for it, and in this way you can definitely win against overwhelming forces with a single well-placed starbase, but a human player will avoid and maneuver around a starbase to make it less effective. It's very difficult to use starbases effectively in multiplayer and you definitely need to rely more on your fleet.
The turret and hangar are woefully overpriced. Not useless, but definitely something to avoid. Use scouts to watch for larger attack forces, and use small groups of mobile defenders to fend off weaker forces. Unsupported siege frigates are so fragile you can actually take them out with your scouts. 2-3 scouts per siege frigate is sufficient.
Obviously to get diplomacy going you're going to need to send out those diplomacy ships. What's the ideal way to do this, one per planet for the guy you're trying to make friends with, several per planet, group them all around his capital or what?
No more than one per planet. It doesn't really matter which planet you send them too, unless you're trying to use one of their special abilities to help out the guy. In that case, read the special ability and target planets that would benefit more from it.
Is it a good idea having multiple capital ships in the same fleet (not sure how experience gets split up)
Experience is shared evenly. If you have three capital ships in the gravity well, each gets 1/3rd experience. Sometimes you want to split them up, sometimes you want to keep them as a single fighting force. It really depends on what you're doing and what comboes your pulling off.
What should a late game 'ultimate fleet' look like, multiple capital ships obviously, plus whatever high end cruiser the race has?
The bomber is the most dangerous combat unit in the late game, but each faction has a capital ship (Kol, Halcyon, and Kortul) that can shut them down, so you need to back them up with firepower. Support cruisers and capital ship special abilities are absolutely huge, and the Advent and Vasari factions both have great superweapons for combat (the TEC's is great for shutting down the enemy economy, but once the opponent compensates for it, it's lackluster).
A TEC player this late in the game is in trouble. The TEC fleet is strongest early-game, and just gets weaker in comparison to the other factions later on. You need to be carrying an advantage from the early game, usually in the form of a bigger fleet or higher-level capital ships. TEC basically is stuck using bombers, javelis LRM, or kodiak heavy cruisers for its damage output, but the problem is the other factions have lots of ways to shut down all of these unit types, so you need to compensate with overwhelming numbers or devious special abilities to counter the enemy. Basically, it's about quantity for TEC, and if you don't have it, you're in big trouble.
Advent will want to build their fleet around the Iconus Guardian. Its two abilities are absolutely critical to keeping your forces intact. Its first ability is called shield projection, which allows the guardian to absorb some of the damage dealt to nearby friendly units. You combine this with the shield regeneration ability of the Progenitor Mothership. The result is massive shield regeneration abilities. This can make the Advent fleet nearly impervious to damage while the combo is active, and if the enemy can't hit with overwhelming power or shut down these abilities, they won't even scratch you. The other ability of the Iconus Guardian is repulsion, an ability that pushes away enemy units. This can be used to keep enemies where you want them and protect your valuable capital ships.
Vasari is the most dangerous faction if they can max out their fleet. And really, it's no one thing, it's everything about this faction that comes together into a beautiful synthesis. The brutal phase missile weapon, the devious stikulus subverter that can leave enemy frigates completely helpless, the Kortul Devastator that can simply disable enemy strike craft weapons, a Skirantra/Overseer combo offering massive healing, putting up minefields and starbases in enemy territory... and let's not forget the antics you can pull with phase stabilizers and kosturas. This is just a faction that has everything going for it late-game. If you get this far as Vasari, you have so many awesome options available.