My car comes with 175/70R13 tires and 13 inch rims. I got a nice set of alloy wheels but they are 14 inch.
My options of tires are 185/60R14 and 175/65R14
For the 185/60R14:
-increase the width of the tires
-keep the circumference almost equal(less than half percent)
For the 175/65R14
-keep the width of the tires
-will increase the circumference by about 1.5%, which will actually make my speedometer almost perfectly accurate(its off by about the same amount of percentage)
What will be the result in terms of gas mileage and acceleration for both of these?
What is the year, make, and model?
If you are going to 14 you should follow the plus one strategy to keep the speedometer correct. Since stock is 175 70 13 , you should go 185 60 14. This continues for every size up , so for a 15 it would be 195 50 15. The bigger wheels/tires will handle better although you will lose a little ride quality due to less sidewall. Heavier wheels/tires can slow acceleration and increase gas use but I doubt you would notice anything.
If both have the same rolling resistance, the wider tire has more friction so the mpg will be slightly lower. About the acceleration , the wider tire gives you more interaction with road and lower the chance of spinning but in general, there should not be any difference unless you always burn out tires while you accelerate.
larger diameter = better fuel economy on highway, poorer in the city. worse acceleration. I would stick to the original wheel size.
Poor man's solution

Going wider will give you better traction but will put more stress on the rolling assembly, especially the front. Going to a larger diameter will make the speedometer read slow and put more stress on your transmission, especially if it's an automatic. Pick Your Poison!
Go with 185/60R14. Your car will handle better, will be more responsive, more communicative and fuel consumption difference will be insignificant.
F.S.
When changing sized from stock. I prefer wider for better handling, but I always check to see how much off the speedo will be.
https://www.sizemytires.com/calculators/compare