10W-30 can be ok if used in really hot climates or if you do a lot of highway driving.
Generally 5W-30 is best, since it's slightly thinner and gives better mileage and can also get into places in the engine that thicker oils don't get to as well.
Also keep in mind there is a lot of viscosity overlap, there are synthetic oils that are labelled as 5W-30, but which are very very thin like a 0W-20, and there are also 10W-30 oils that are thin and closer to that 5W-30 viscosity, so it's not like every oil can be judged based on the label alone.
In general, a top brand name 5W-30 oil is what you want. If you go with a synthetic, a 5W-30 or even a 10W-30 can be fine.
If you think the slow starting is due to the oil, and not just a battery/starter or electrical issue, you could switch to 5W-30 oil on the next oil change and see if that fixes it. Be sure to just use 5 quarts, as overfilling the crankcase can have the effect of slow starting, since the crank will have to turn through an extra amount of oil to get moving, and this will also effect mileage.