The process for an agency-level protest involves filing a formal protest with the contracting agency, and the agency is required to make its best efforts to resolve it within 35 days. An agency-level protest based on an apparent solicitation impropriety must be filed before bid opening or the closing date for proposals. In all other cases, it must be filed no later than 10 days after the basis of protest is known or should have been known, whichever is earlier, as stated in FAR 33.103(e). Since your concern is with the evaluation, your protest would fall under the "all other cases" category, and you would need to file within 10 days of when you knew about the alleged evaluation issues. This means acting quickly after receiving the award notice.
For federal contracts, the protest procedures are outlined in FAR Part 33. It is standard practice to request a debrief first, as it can provide valuable information about the evaluation and help you decide whether to protest. You are entitled to a postaward debriefing if your written request is received by the agency within 3 days after the date you received notification of contract award (FAR 15.506(a)(1)). For state and local procurements, the process and timelines will be found in the specific state or local procurement code and the relevant state procurement portal's vendor guide.