Amniota: Coordinated evolution in the conquest of the land:

Reptiliomorpha - Closer to Amniota:

Caerorhachis bairdi (Carboniferous) a basal reptiliomorph from Wikipedia
When we first encounter the amphibian and reptiliomorph lineages in the Early Carboniferous, their members don't look that different, and probably weren't dissimilar ecologically. Even so, even basal members like Caerorhachis (Early Carboniferous - right) possess evolutionary novelties that characterize Reptiliomorpha:

Like basal amphibians, basal reptiliomorphs show varying degrees of adaptation for life on land, and a range of adaptations, but none are absolutely independent of water. Some interesting highlights that are successively closer to crown-group Amniota:

Discomfort and forebodings: Despite previous claims, some of the physical characteristics of Lepospondyli are potential synapomorphies with Lissamphibia! Indeed, although our consensus cladogram shows us getting very close to Amniota when we discuss lepospondyls, their likely ecologies are disturbingly amphibian-like. As a result, although a solid consensus exists about the non-lissamphibian tree, Lissamphibia are significant wild-cards. For the record, three views exist about Lissamphibian phylogeny:

There is no firm professional consensus, although the recent redescription of the superficially caecilian-like ?Rhynchonkos stovali by Szostakiwskyj et al., 2015 raises doubt about the caecilian-microsaur link. Stay tuned.

The terrestrial enigma: Although we have climbed closer and closer to the last common ancestor of Amniota - the true land vertebrates, we see no real trend toward terrestriality. Indeed, Seymouriamorphs, the most terrestrial of the lot as adults, definitely had water-breathing larvae. What are the evolutionary dynamics of this transition? First, consider the end point:

Amniota: (Carboniferous - Recent) The most recent common ancestor of mammals and birds and all of its descendants. Amniotes represent a major departure in the evolution of full terrestriality in vertebrates, and are the first vertebrates fully to break the dependance on bodies of water.

life without gills:

As we've seen, the presence of gills is a positive liability for any animal seeking to spend time on land. With the rise of Amniota, we are definitely dealing with animals that do not use gills at any life stage. Their loss entailed a new set of problems.

Two-chambered hearts:

Fish-like vertbrates have a straightforward cirulatory system in which blood:

Three-chambered hearts:

Among lungfish and tetrapods, for whom the lungs are an important source of oxygen, a set of blood vessels have become specialized to take blood from the ventricle directly to the lung, then back to the atrium. In lungfish, the atrium is partly partitioned. In tetrapods, it is completely divided into a left and right atrium, receiving oxygenated blood from the lungs and deoxygenated blood from the body respectively. This blood gets mixed in the ventricle, but this is not a problem in lungfish (or any ancestral stegocephalian that relied to any degree on gills), since its next stop is the gills, anyway. For exclusively air-breathing tetrapods, however, it is a big problem, because it means that blood reaching the body is usually not completely oxygenated.

Four-chambered hearts:

Among some amniotes, this problem is solved by the evolution of a heart in which the ventricle, in addition to the atrium, is completely divided. Thus, deoxygenated blood follows this course:

The four-chambered heart has evolved twice among amniotes:

The common ancestor of amniotes is unknown. The earliest fossil amniote, Hylomonus dates from the Late Carboniferous (312 mya). From the Carboniferous and Permian we have fossils of candidates for its sister-taxon, including:
Looking farther back in time to the Carboniferous, we see:

Westlothiana from

And note that Casineria and Westlothiana are both in the size range that our speculation on the origin of the amniotic egg leads us to expect.