I think your strategy when you play multiple tables should be the same as the strategy when you play one table.
This is actually incorrect. When you move up the amount of tables you play, you will actually need to find shortcuts or "default" plays. So my strategy changes a lot when i multitable, i'm looking to make decisions easier so i don't have to think about them as much and thus, spend less time thinking about them. But when playing few tables, i can think harder about what the most optimum play is and of course take much more time. Since we only really have 30 seconds on a single decision (or whatever the site allows) we simply cannot play the same way as we would with fewer tables.
It depends how many tables you are planning on playing, if its only a few more than normal (1) then you should probably be playing the same as you normally would but if its a lot more, eg. 12 - 15 + then you need to be tightening your range imo as you will be getting more hands more frequently
If you are tightening your range with playing more tables, you shouldn't be playing so many. This is obviously something you don't want to do that i mentioned above about making decisions easier for yourself.
So how do you make decisions quicker and easier without sacrificing quality of your play?
Experience is probably the most important part. You don't start out playing 24 tables and expect to be able to play them profitably. Start out slow, and work up. You can use trackers to find out how you are doing when playing "x" amounts of tables. I started out like you, with 1, moved slowly onto 2, then onto 4. And by slowly, i mean playing 100-1000 games or hundreds of thousands of hands in cash until moving up in tables. The amount of games/hands it takes will vary for each individual. Some may pick it up quicker, others slower. The easy rule is play as many as you feel comfortable with, then push yourself to move up.
Secondly, review your games! this is also huge. You will find in pretty much any form of poker that you will come accross very similar decisions as you play more. So pick up on these decisions and simplify them. you can use review to find out and study what types of ranges are profitable and then find ways to kind of find shortcuts to use while in game.
Software is good, but really only when moving up from say 12 tables to 15 or something. It will buy you that little bit extra time which is crucial sometimes. Table ninja is good but expensive, hjalper is the same. I believe most
poker sites actually have hotkey setups, so do some research and spend some time finding out a good setup that you like. This includes the layout, color, key placement, computer setup, computer speed, internet speed etc etc.
i've just lost my train of thought, i might be back later to post some more.
This is also written from a SNG perspective, i've never really multitabled cash games, but the theories are still the same i'd say, just with less amounts of shortcuts while playing, and likely less tables/differant setups.