The top left photo appears to be Bermuda grass which is a good semi-drought tolerant grass. The bottom left could be St Augustine which requires more water, and can be damaged in extreme winters.The 2 on the right we are assuming are of your yard. It is very difficult to positively ID them. But we would like to make you aware that trying to grow grass under large shade trees is difficult if not impossible without doing damage or harm to the trees themselves. It’s not just the shade that makes it difficult, it’s the feeder roots of the trees, these are the part of a trees rootstock that spreads out horizontally just under the soils surface and extend out to the edges of the tree canopy. As a tree ages and gets broader so do the feeder roots below. This broadening begins to compress or ‘squeeze ‘ the soil between the roots. This heavy compression of the soil is too hard packed to allow most things to grow under it. In the natural world this is a trees defense, disallowing things to grow under it also means it’s not in competition for food or water. We would highly recommend you speak to a licensed arborist before attempting to lay a sprinkler system or install a lawn near your trees. As far as making a good grass choice you should contact your local extension service and find out what they would recommend for your area and soil type.