SDS-PAGE is the normal way, at least for an initial test expression. For a Western blot or ELISA you need antibodies specific to your protein, which you probably don’t have. An easier way to identify, for example, would be to cut the SDS-PAGE band and perform mass spectrometry on it to try to elucidate its protein sequence.
To follow on, typically proteins produced this way have a tag, or handle, to specifically pull it out from all the other proteins. A typica tag is 6 x Histidine amino acids at the beginning of the protein. These histidines bind to nickel, which is immobilized on a resin. You can use antibodies against the 6xHis to Western for your protein of desire without needing a protein specific antibody.