There are two main gravitational causes of tides: the Moon, and to a lesser extent the Sun.
When the moon is full or the moon is new, the Earth, Moon and Sun are roughly aligned, and the Lunar tide combines with the Solar tide to give a "Spring tide" that has a larger range.
When the moon is at first quarter, or third quarter (ie a half moon), the solar tide and the lunar tide are acting against each other, and you get a "neap tide" with a lower range than a spring tide.
The size of the tide on a particular day at a particular location is strongly affected by the local shape of the coast and sea floor, Some places only have one tide each day, or tides of different size, due to these effects.
Note the name "Spring tide" is not from the season. You get Spring tides every New and every Full moon. It comes from an old English word meaning "to bulge" The tide "bulges" during New and Full moon.
So you are not correct to say that tides are higher on the day of Full moon. The New Moon's gravity does create tides, and tides at New moon are as large as the tides at Full moon, and larger than the tides at half moon.