Unraveling this mess may be difficult. Seeing the 30 amp breaker join the two sides indicates that the two hot lines go straight down each side in the back of the panel rather than alternating every other breaker. If it were alternating, then any 220 circuit only needs two vertical slots rather than taking up all the space of both sides as well. What this means is that these two circuits can be joined anywhere since the lines are in phase with each other.
The easy fix is if you have an incorrect connection in that breaker panel, rather than each line going its separate way. If that's the case, you remove the connection and the problem is solved. So as others say, the first step is the open it up and take a photo (with the power off and being careful to avoid the lines feeding the main breaker).
The difficult fix is if the connection is somewhere else in the house. Unless you know how the home is wired, you are left with breaking the connection at various points until you find the location that roughly splits the circuit in half. To do this, you open up an outlet in one of the rooms, disconnect the load side of the outlet (pick a side if you don't know which is which), and see if only half the outlets turn on when you reset one breaker. Try to make the split in the circuit between two rooms.
Once you decide where to split the circuit, you're left with an extra wire from the other circuit in the J box to cap or remove. The proper solution would be to remove it, but doing so will likely require opening up the wall. If you cap it, make sure each wire cannot touch anything else to avoid the risk of a short, and then label the wire so no one hooks it back up in the future. I would also go through the extra effort to find the other side of this connection and disconnect it and label it on both sides.
Edit: Seeing the separate red and black wires makes me wonder if they continue through each outlet with two circuits like mgb suggests. If so you just need to locate where the two lines cross. If the two circuits are separated in the outlets themselves, then it should be obvious when you open an outlet up and see a red and black going to the separated top and bottom. In that case, turn the power off to both circuits, disconnect an outlet, cover the exposed connections, and turn the power on to only one circuit. If you now have two separate circuits, then then crossed connection is in the outlet you disconnected or one of the dead outlets in the home. If the circuits are still joined, then the problem is in one of the live outlets. Use the process of elimination to avoid opening too many outlets.
As always, when working with electricity, ensure the power is off before you risk anything touching a wire. Test ever wire in the outlet before beginning work, especially in your situation. And when you open the panel, the line side of the main breaker is still hot even when that breaker is shutoff. If you feel nervous about doing any of this yourself, then hire an electrician.
Every switch sits in-between two hot wires. One is always hot, from the mains, and one is hot when the switch is active, and that one goes to the controlled device(s).
It sounds like you have connected the hot wire (black) from the mains directly to the hot going to the fixtures. The always hot switch should be connected to one black wire on the dimmer and the black from the fixture to the other black on the dimmer. Check the dimmer to see if there is a marking to indicate which is which.
The easiest way to test which black wires are which is to use a non-contact tester. Turn off the breaker. Separate the black wires, put a wire nut on the exposed end of each and turn the power back on. Check each black wire to see if it registers as hot on the tester. The one that is hot is the mains. The one (or more) that are not go to the fixtures.
In general, all white (neutral) wires are connected. This assumes the white wire is not being used as a hot line. If it is, it should be marked with a black marker or a piece of black tape. Similarly all greens and bare wires (ground) are connected.
Some dimmers need a neutral wire and some do not. It sounds like yours does. It should be connected to the other neutrals. Similarly the grounds should all be connected.
Best Answer
It sounds like you a ground loop issue. Take a look at "ground loop isolators" on amazon, best buy, etc... They might help the issue. This happens to me sometimes when I am putting a heavy load on a circuit in my house. (Vacuums, Hair Dryers, etc will do this)