Because of the nature of this game, this is a separate file with information that can potentially spoil game surprises or otherwise give away more information than my usual set of hints. There is a normal review if you are thinking about buying the game.
If these tips don't include the information that you need, try the Evil Genius Forums for more information.
Make the various rooms fairly large. Much of the equipment needs a lot of space (including space around it to access). Be careful not to create little 'dead' zones in the rooms. These are small areas that the equipment layout that a minion cannot get into (or out of).
Security cameras should cover the entire base. Agents can sometimes dig into the 'dead' zones of your base, and security cameras will be your first point of protection.
Minions need locker space far far more than they need beds. To expand your number of minions, first add more lockers. Also there is an ultimate limit of 100 minions, to change that, you'll need to engage in some cheating. See the population tip in the Evil Genius forums. (Note, while some purists claim that the game is made too easy by this cheat, I feel that 100 minions makes it just a bit too difficult. The only cheat I've used in my game is to increase the minions to a max of 200.)
Henchmen are the key to Evil Genius. Your evil genius is very limited in what they can do. You can control numbers but not actions of the minions. But your henchmen you can control and send off to missions or attack intruders. They can fight off any number of soldiers or other enemies. The only vunerability is to the super agents. They may faint after being overwhelmed by soldiers, or have to return from the world map after a defeat there. But watch them around super agents because they only have 3 lives and they are the most useful tool you have.
While the description of the available henchmen gives you some idea of their quality, since you may only ever have seven at a time, you want to choose the correct seven. This FAQ of Evil Genius has a list of all the henchmen and their abilities. To jump down to the right place use control-F (Find) and the phrase to search for is QFH.
You must eventually leave the first island. If you have started on the cursed totem Act of Infamy, you should finish it because that sub-objective will disappear once you move to the second island.
Destroy as much as you can before you leave the island. In particular destroy everything in the research lab, as many of your power generators as you can, all hotel stuff, etc. Why? Well, the only things your minions take are the loot items you have acquired. And destroying an item will give you back some of the purchase price. It takes much cash to build the new base, so the more you can get from the remains of your old base the better. Save the game just before your Evil Genius leaves the island.
On the second island, you'll be quite vulnerable until you can build enough of your new base. I ended up moving the loot around so that minions could surround it (thereby protecting it from thieves) and simply start the new base with a very long corridor and a door. All the loot was moved to the corridor as quickly as possible. I did have to reload the game once because a thief got something and got away before anyone could catch him.
When, where and how many traps you lay around is a personal preference. Better traps can net you cash (when an agent hits two or more traps at a time, you gain money), but there is the important minion question. Namely, is this going to do too much damage to minions?
My first traps were just inside the base. This worked ok, until after a mission that pushed my heat up too far. Then I had lots and lots of enemies on the island, too many entering the base and losing tons of my minions to the traps triggered during the fighting. Then I discovered the beauty of using huts with locks that keep the minions out and multiple traps within. Beware the agent caught in these beauties.