Do you need to cut any counter sections to length? That's the hardest cut to make on post formed counters (the kind you get from Home Depot or Lowes and already has the backsplash attached and often have rounded edges). The ones in your kitchen appear to be this type. Cutting to length is best done on a table saw with a good, sharp, carbide toothed blade. Pay the few extra bucks, measure very carefully, and have the lumber yard do that cut for you.
Then there are mitre cuts. Fortunately, with premade counters, you won't have to do them.
The last cut is for the sink. And that's the easiest. I'm sure you'll be reusing your 'new' cast iron sink. Remove it from the old counter and measure the hole carefully. Copy those measurements to the new counter. Lay some blue painters tape or masking tape over the line of the cut and then re-mark the cut lines. The tape will help prevent chipping of the laminate. A little chipping or a slightly 'off' cut is likely not to be catastrophic, as the sink will cover the raw edge of the cut.
When you're ready to make the cut, use a sabre saw or a skill saw.
Skill saw:
Sabre saw:
The method differs, depending on the saw you use. The sabre saw is the most forgiving, but the skill saw gives the better result.
Let's assume you'll use the sabre saw. Drill a hole inside the cut line (in that portion of the counter that will be the scrap, where the sink will be). Locate the hole near the cut line, but on the inside of it and make the hole big enough for the saw blade to fit into it. Put the blade into the hole and start cutting. When you get to corners, you can either try to cut them to a tight radius, back the saw up a bit and cut diagonally (you'll remove the uncut section later) or you can drill a new hole and repeat the process for each side. If you do the diagonal cut, you can go back from the opposite direction after you get the rest of that cut done. Or you can start by drilling one hole right at each corner and avoid all the corner cutting headaches entirely.
Be careful about the scrap dropping out. You can have a helper hold the piece and catch it before it drops. A one-person method is to screw a board to the top of the scrap. The board should be long enough to bridge the entire sink cutout. The board will hold the scrap and stop it from falling. If you do this, cut two parallel sides, screw the board to cross the cuts, and then do the remaining two parallel cuts.
If you use the skill saw, you start the cut by doing what's called a 'plunge cut'. You set the front of the saw's shoe (sole, bottom) on the counter. Line up the blade with the cut. The blade guard has a small tab at the bottom rear. Use this to move the blade guard clear. Start the saw and slowly lower the blade into the cut. It will cut as you lower it until it is flat on the counter. Then just push it to the end of the cut.
Note that a circular saw, unlike a skill saw, will **not** cut all the way into the corner. The top of the cut will get to the corner, but, given the blade's radius, the bottom of the cut will be a bit short. This actually works to your advantage here as the uncut section will hold the scrap in place. Make cuts on the two parallel sides, screw the safety board in place and then do the last two cuts. Even though the uncut sections should hold the scrap and prevent it falling, use the board too, just for safety.
Finish the cuts with a hand saw or ... tada .... a sabre saw.
It is actually easiest to do this cutting with the countertop in place. Just be careful that the falling scrap doesn't damage the floors of your cabinets.
Good luck!