I read this intetesting page about fit indices in SEM. There are lot. It seems that the RMSEA is one you should always report, but how to choose the other ones? Is there some kind of rule of thumb? Cherrypicking is something I want avoid.
The go-to classic reference for this matter is Hu & Bentler's (1999) paper on fit indexes for SEM. In the paper, they advocate for a "2-index presentation strategy", for which you report an absolute index of model fit, and a relative index of model fit.
Absolute indexes of model fit compare the fit of your model against a perfect fitting model. Recommended absolute indexes include the RMSEA (especially nice, because you can get a confidence interval to perform a test of close fit), and the SRMR. The recommended cutoffs for these two are generally similar; the smaller the better, and < .08 is generally recommended to be considered acceptable, though these guidelines may vary by discipline.
Relative indexes of model fit compare the fit of your model against a "null" model--a deliberately poorly fitting model that is still reasonable (usually that all observed variables are uncorrelated with one another). Recommended relative indexes include the CFI, and the TLI/NNFI (different acronyms, but the same index). Again, the recommended cutoffs for the CFI and TLI/NNI are generally similar; the bigger the better, and > .90 is generally recommended to be considered acceptable.
Note that your model fit index selection and/or calculation might need to change depending on your modelling needs. For example, Little (2013) suggests that you should use a manually amended null model when calculating relative indexes of model fit for longitudinal structural equation modelling. And when comparing groups for measurement purposes, the change in CFI between competing models can be a particularly useful index (Cheung & Rensvold, 2002).
Cheung, G. W., & Rensvold, R. B. (2002). Evaluating Goodness-of-Fit Indexes for Testing Measurement Invariance. Structural Equation Modeling, 9, 233-255.
Hu, L., Bentler, P. M. (1999). Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives. Structural Equation Modeling, 6, 1-55.
Little, T. D. (2013). Longitudinal structural equation modeling. New York, NY: Guilford Press.