How do I hook up old DVD player to new HDMI TV

That's the best route. Otherwise, a Component to HDMI converter box would cost the same or more than just buying a basic Bluray Player. And you'd be able to play DVDs and Bluray HDs, both of which would already be digital video/audio. The box would have to convert analog component video and audio signals to digital for HDMI. Likely resulting in an picture lacking in overall sharpness.