Update readme

Author: charleskawczynski

README.md

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 # ClimaTimeSteppers.jl
 
-[![Docs latest](https://img.shields.io/badge/docs-latest-blue.svg)](https://clima.github.io/ClimaTimeSteppers.jl/dev/)
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+| **Docs Build**       | [![docs build][docs-bld-img]][docs-bld-url]   |
+| **Documentation**    | [![dev][docs-dev-img]][docs-dev-url]          |
+| **GHA CI**           | [![gha ci][gha-ci-img]][gha-ci-url]           |
+| **Code Coverage**    | [![codecov][codecov-img]][codecov-url]        |
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+[docs-bld-img]: https://github.com/CliMA/ClimaTimeSteppers.jl/actions/workflows/docs.yml/badge.svg
+[docs-bld-url]: https://github.com/CliMA/ClimaTimeSteppers.jl/actions/workflows/docs.yml
+
+[docs-dev-img]: https://img.shields.io/badge/docs-dev-blue.svg
+[docs-dev-url]: https://CliMA.github.io/ClimaTimeSteppers.jl/dev/
+
+[gha-ci-img]: https://github.com/CliMA/ClimaTimeSteppers.jl/actions/workflows/ci.yml/badge.svg
+[gha-ci-url]: https://github.com/CliMA/ClimaTimeSteppers.jl/actions/workflows/ci.yml
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+[codecov-img]: https://codecov.io/gh/CliMA/ClimaTimeSteppers.jl/branch/main/graph/badge.svg
+[codecov-url]: https://codecov.io/gh/CliMA/ClimaTimeSteppers.jl
 
 A CPU- and GPU-friendly package for solving ordinary differential equations