I respectfully disagree with advice to warm lye solution. Don't do that if only for safety's sake. The less fiddling one does with lye solution, the better. If you spill fat, you have a huge mess. If you spill lye solution, someone could go to the hospital.
On top of that -- it's not the temps of the individual ingredients that matter to the saponification reaction. It's the initial temperature of the soap batter that's the key. If you want your initial batter temperature to be warmer, heat the fats rather than the lye solution -- much safer and just as effective.
If you are soaping with a moderate to high percentage of solid fats, IMO the fats should be just warm enough to be fully melted and clear. I make recipes high in lard. When using 50% lye at room temp and my fats are just melted and clear, I normally see an initial batter temperature of 95-105 F (35-40 C). If soaping with all or mostly liquid fats, then room temperature is fine.
You'll see a slight rise in temp, even if starting with room temp ingredients, because there is some heat released when the 50% lye mixes with the rest of the water-based liquids.
As you get more experience with this, you'll probably adjust the temps to work best with your particular style of soap making, but I hope these guidelines will help you feel more confident.