I’m looking for any recommendations on how to convert monsters in adventures from legacy rules to current rules.

I’m doing my best, but I’m a new DM so I’m open to suggestions.

For reference I’m planning on running Lost Mines of Phandelver for a group of friends. I’d like to use the core rules, but the monsters in the guide are on the legacy rules.

Use this to calculate how much to up their stats: 2024 Monster Manual on a Business Card « Blog of Holding

Outside of stats, there are a few changes made to monster design philosophy:

• Attacks do not require both an attack roll and saving throw
• Monsters can do more damage dice than normal with their weapons
• Legendary monsters get more LAs and legendary resistances in their lair
• All legendary actions cost 1 action, some can only be used once per round
• Resistance to non-magical BPS is no longer a thing. Instead, some monsters are resistant to all BPS damage
• No monsters use spell slots. Instead, they get a smaller number of spells that can be cast either at will, 3 times per day, or 1 time a day. occasionally a spell can be cast twice a day.
• Most monsters have better action economy. For example, monsters that can swallow can do so as a bonus action
• Swallowing is now a saving throw, rather than an attack roll
• Constrict/grapple type attacks are typically Strength saves, not attack rolls
• Most monsters with multiple attacks that are basically the same (Like a claw and bite multiattack) Now only have one attack for simplicity. 
• Fire Aura type effects trigger at the end of the monster's turn, rather than when the creature is hit by a melee attack.

I think that's it for the major changes.

So should I just keep the monsters or continue with near equivalents of the same CR? Like bugbears (legacy) = bugbear warrior (current)

You'll probably want to modify the stat block according to my above post if it's adventure specific. If it's not, use the official version.