In Red Alert 1, multiplayer strategy mostly —and unfortunately— consists of building a small powerful attack force as quickly as you can and then immediately launching an offensive at the enemy in an attempt to destroy key structures in his base before he is sufficiently prepared to defend them. Typically this is known as 'rushing' and the most popular units to use in a rush are tanks, hence 'tank rush'. And of course the most popular structure to target using a tank rush is the opponent's construction yard. When a tank rush is successful —and especially so against inexperienced players— it will more often than not result in a 'psychological victory' where the opponent is, for all intents and purposes, not actually defeated but his morale is so crushed from losing his construction yard that he concedes without a battle.
More experienced players will however immediately launch a counter attack at the rushing opponent knowing that they may have gambled everything on the success of the rush and did not sufficiently prepare for (or even expect) any form of retaliation.
Unfortunately, in my experience, multiplayer does not often get much more sophisticated than this. Forget about building a nice pretty walled-in base with a bunch of perfectly placed Tesla coils and flame towers and then reconning and planning a sophisticated offensive strategy with a series of complex flanking attacks. It just ain't gonna happen.
And this is simply because there is no time. For the most part, fully expect your human opponent(s) to —in very short order— come charging into your base with a platoon of heavy tanks, stop for a brief second to laugh at your Tesla coils and then proceed to lay waste to everything in sight. Basically the sum total of your multiplayer strategy will be GET THEM BEFORE THEY GET YOU. Period.
Here are some good offensive and defensive multiplayer strategies I've seen work, starting off with the quintessential...
Widely used (and overused), this strategy is a no-brainer. Simply pump out six tanks as fast as you can and then send them immediately to the enemy base. Race them right past whatever base defenses the opponent has up and running and go straight for his constuction yard. Don't bother fighting off anything that attacks your tanks. This is a throwaway force. It's job is to destroy the enemy construction yard, do or die.
Play as the Allies and build light tanks for the fastest rush. Play as Soviets and build heavy tanks for an extra devastating attack (e.g., take down more buildings after destroying the construction yard). Or you could go with five Soviet heavy tanks instead of six to cut down on build time.
Against more experienced players, be ready for a counter attack. Executing your rush means time away from building up your base defenses (unless you're really good). Experienced players who are not intimidated by losing their construction yard know this and will launch a counter attack immediately after your tank rush knowing that your base is weak and vulnerable. (Construction yard? Construction yard?! We don't need no stinkin' construction yard!!)
This is where you play as Soviets and you put all your resources into building six airfields and six MiG aircraft. Basically, it's a bluff because it's costly and time consuming. To pull it off you have to go very light on base defenses (which means few or no tanks) in order to get your MiG attack force up and running as quickly as possible. But once you they are up and running (and you've managed to recon your opponent's base with some infantry), it's simply a matter of sending all your MiGs to attack the opponent's construction yard. This will usually take it down in one pass and if the opponent has few or no air defenses, it will almost always get him to concede. Just the prospect of you returning every 60 seconds to completely obliterate yet another key structure with your near invincible lightning quick MiG attack force is enough to get most to throw in the towel.
An excellent strategy to use along with the MiG rush plan is to send out a couple or three tanks ahead of time to mill about near the opponent's base perhaps taking potshots at his harvester or something like that. This will keep him on edge thinking you're about to launch a devastating tank rush any moment which will buy you enough time to build up your MiG force.
I've managed to pull this off a number of times but usually it's limited to only once per opponent because it's easy to defend against and most players will wise up after the first time (especially because it's so humiliating). To pull this off, you play as the Allies and then quickly build up an offensive force consisting of one APC, 3 rocket soldiers and two engineers. Load everybody into the APC and send it racing into the opponent's base (stealthily if you can) and park it right beside his construction yard. Then quickly unload everybody, send the two engineers into the construction yard to damage it and then have the rocket soldiers attack the construction yard to finish it off. Usually 3-4 shots from all your rocket soldiers will be enough to destroy the construction yard after damaging it with the engineers. You can use the APC to either ward off any infantry or attack dogs he might send against you or to help attack the construction yard.
This attack plan is faster and cheaper than the ever popular Allied 6-light-tank rush but admittedly much more precarious (but it's a hell of a lot of fun if you can pull it off). Make sure you set everybody as teams beforehand in order to exercise quick control over your squad once they unload from the APC inside the enemy base (e.g. set the engineers as one team, set the rocket soldiers as another team, set the APC as a third team). Once you're in the thick of the fray, you don't have time to be fumbling about trying separate your engineers from your rocket soldiers.
This attack plan can be made even better by using a diversionary force. Build two light tanks along with the infantry/APC squad. Then when you're ready, set the tanks as a team and then load up your APC and send everybody out toward the enemy base. Halfway across the field, split off the tanks and have them go harass the enemy harvester. This will usually panic your opponent into thinking you're tank rushing his harvester after which he'll send all the units in his base out to defend it. This will then be a perfect time to send the APC engineer/rocket soldier squad into his base to take care of business.
I haven't seen many defensive structure strategies work in mulitplayer because it's usually all about rushing and counter-attacks which invariably means spending all your resources on mobile units that can both defend and then quickly go on the offensive. However, one defensive strategy involving structures that I've seen work quite well —and was incredibly annoying— was pillbox spam. This is where you play as the Allies and then you liberally, and I mean LIBERALLY, pepper pillboxes willy-nilly all around your base. They're relatively cheap, they build quickly and can actually do a very passable job of chewing up a tank rush.