You do not have the right to speak to HR. HR is not your friend, don't treat them like one.
HR is here to protect the company, not you. They will happily act like your friend, write down everything you say, and then report it to your old manager. Are you still excited to talk to HR?
My former manager is putting pressure on my current manager to get me
fired.
If you have any proof of this make sure to make a copy of it and keep it somewhere safe, and offsite
I tried to go to HR. HR refuses to meet me
Save any and all correspondence with HR. Including that you've completed coaching programs somewhere safe, and offsite
Why is your previous manager working so hard to get you fired? Can you get any communication trails between your old and new manager? If you can keep them somewhere safe, and offsite
Do you believe your old manager is retaliating against you? If you suspect so, you should contact HR 1 more time. Do the following
BCC your personal email address. Make sure you have access to this email trail offsite.
Say you suspect your old manager of retaliation. Be sure to use the word retaliation. Any half-decent HR department will take notice (in the US and the UK). The company as a whole could be liable if they allow the old manager to continue retaliating against you.
Save what HR says somewhere safe, offsite
While not part of the original question, it's also likely time to start looking for a new job.
You may not succeed in getting protection from HR. Even if you succeed, you've likely been taken out of the running for raises, promotions, and bonuses for the year due to the back-and-forth.
HR being involved is usually a sign to get out. Remember HR protects the company's interest, not yours. That's usually seen as management interest. Use this to negotiate a severance package once you quit.