Hit Points and Hit DiceYou gain the Hit Points from your new class as described for levels after 1st. You gain the 1st-level Hit Points for a class only when you are a 1st-level character.
You add together the Hit Dice granted by all your Classes to form your pool of Hit Dice. If the Hit Dice are the same die type, you can simply pool them together. For example, both the Fighter and the Paladin have a d10, so if you are a Paladin 5/fighter 5, you have ten d10 Hit Dice. If your Classes give you Hit Dice of different types, keep track of them separately. If you are a Paladin 5/cleric 5, for example, you have five d10 Hit Dice and five d8 Hit Dice.
Your Proficiency Bonus is always based on your total character level, as shown in the Character Advancement table, not your level in a particular class. For example, if you are a Fighter 3/rogue 2, you have the Proficiency Bonus of a 5th- level character, which is +3.