Dinosaurs and the ancestor of birds are both archosaurs, so they are more closely related to each other than either of them are to lizards. Lizards are lepidosaurs. Archosaurs are a bit more "advanced" or derived than lizards. However, exactly how close dinosaurs are to birds is still being debated, although the cladists want the debate to stop because they think that it is already set in stone, and that it is indisputable gospel truth that birds can only have evolved from a dinosaur. The dinosaurs are extinct, as are most other kinds of archosaurs, and the only surviving archosaurs are crocodilians. We do not know if birds are more closely related to crocs or if crocs and dinos are more closely related to each other than they are to birds. Ancient birds have crocodilian teeth but some dinosaurs (e.g. the theropod Scipionyx) may have a diaphragm, and so do living crocodilians.
The only known reptile with feathers is Longisquama insignis from the Triassic of Central Asia. So, it is most likely that birds evolved from the basal archosaurian Longisquama or one of its closest relatives, but crocodilians and dinosaurs are closer relatives of each other than they are to birds because of the diaphragm they have in common. Birds do not have diaphragms, only air sacs.
As for dinosaurian feathers, it is a hoax. The first supposed feathered dinosaur is Sinosauropteryx of China, which is a close relative of Compsognathus of Sohnhofen, Germany. Sohnhofen preserved many feathers and Archaeopteryx, and yet there is no trace of feathers on Compsognathus. It would be very odd that Sinosauropteryx is feathered but its close relative Compsognathus is not. At the end of the 20th century, a group of 4 experienced scientists (John Ostrom, Peter Wellnhofer, Alan Brush, and Larry Martin) went to China to look at the supposed "feathered dinosaur" Sinosauropteryx. When they returned, not one of them would say that what they saw was feathered, not one. And yet you will find no shortage of web sites or scientists who claim that Sinosauropteryx is a feathered dinosaur. Other scientists have published papers claiming that what they find on Sinosauropteryx and other dinosaurs are most likely collagen fibers, which is an ancient protein that can be found on the skin of many animals, including animals as primitive as a nematode worm. Yet people who firmly believe that Sinosauropteryx is a feathered dinosaur are not persuaded by the lack of evidence of Compsognathus feathers and the lack of evidence that Sinosaurotperyx has feathers.
There is also a second group of fossils that are claimed to be feathered dinosaurs, such as Microratpor, Anchiornis, Protarchaeopteryx, Caudipteryx, and even Archaeopteryx. The problem with these claims is that these animals also have a lot of birdy traits, in addition to feathers. Therefore many scientists claim that they are birds, not feathered dinosaurs. In conclusion, some scientists who desperately want to hang on to the belief that birds are descendants of dinosaurs (primarily because they don't want to suffer career damage by admitting they were wrong) believe that Sinosauropteryx has feathers and that Microraptor and birds that do have feathers are dinosaurs. In short, if the evidence does not fit the theory that birds evolved from a dinosaur, then make it fit.
