If I’m legally registered as a resident in Spain, will I enjoy further free movement rights within the rest of the EU after the end of the transition period?

No. Being registered as a resident in Spain will let you live from 5 to 10 years uninterruptedly, depending on the type of residency you have. However, this residency won't grant you these rights in other EU states. For the rest of the Schengen area or EEA you will be allowed entry for 90 days every 180-day period. Surpassing the allowed period might incur into a travel ban.