Western blot is the most simple technique I think, possible to do without radio labelling with caged phosphates now. As phosphorilation is reasonable to do with quantification more simple method could be gaz chromatography analisys, only it is not available in every laboratory. The alternatives are radiography with P32, MS, FRET...

There are lots of commercial high-throughput assay technologies for measuring phosphorylation of peptides by kinases. The same technologies could, in principle, also be used for proteins, but would either require a lot of protein or a modified protein.

Example 1. ADP-Glo (Promega). A luciferase-based ADP detection technology could be used to detect the ADP product of the kinase reaction, using your protein as the acceptor. A good measurement requires that at least a few % of the ATP gets converted to ADP, so to keep the amount of protein needed to a reasonable level, you have to keep the ATP concentration pretty low. Also, since this is a luminescence measurement, you can do the reaction in a low-volume 384-well plate to conserve material. You need a luminescence plate reader. The ADP-Glo reagents are pretty expensive. There are also other commercial reagents for sensitive ADP detection. I think this would be the simplest approach to implement if you have plenty of acceptor protein to work with and both the kinase and the acceptor protein are highly purified,to avoid background ATP hydrolysis.

Example 2. TR-FRET for a tyrosine kinase. The FRET donor in this case is a lanthanide cryptand, and the acceptor is an organic dye. Both need to be bound at the same time to the protein to get energy transfer. You can buy an anti-phosphotyrosine antibody with a lanthanide cryptand attached that will bind to a His-tag on a protein. The protein has to be covalently modified with a suitable acceptor fluorophore. This method could use very little protein and allows any ATP concentration to be used. It requires a plate reader capable of time-resolved FRET measurements. It's more complicated than example 1, but you don't need a purified kinase to use it.

you could always try confocal microscopy but this depends upon the antibody that you are using and whether or not it has been tested for this application.  I have performed confocal microscopy on phospho proteins to validate my western blot data.  If you do try confocal microscopy, you do not want to use milk as your blocking buffer because it contains phosphotases.  We have used the odyssey blocking buffer from licor.

No easy method to detect protein phosphorylation because the incorporated phosphate is a tiny amount. Immunoprecipitation by a specific antibody,  western blotting and luceferase - based ADP detection are available methods as explained by the other anssers. 

You may want to try two dimensional gel electrophoresis. You would first have to separate the protein by molecular weight by regular gel electrphoresis then do a second dimension run using isoelectric focusing which would separate the unphosphorylated and phosphorylated form of the protein according to their isoelectric points. The proteins will migrate and once they reach their IEP the migration will cease. Since the unphosphorylated and P-forms have different IEP then they should separate. You can then stain with a dye like coommassie blue to visualize.