If I call a company and their recording informs me "the call may be monitored for training purposes" can I legally begin to record that call myself?

As they inform you before recording, you can also record in agreement with them

This depends for what purpose you want to record the call. In case that you want to have a prove, you can easily take the name of the agent and time and date of the call to follow up because it's already recorded and you have the right to call again if there is any problem asking for the supervisor/manager and give him call details and he will get back to you. Some call centers systems automatically send a summery for each contact to assure the customer that his request is under processing by email or SMS. But in case that you want to use the recording in courts or for legal purpose you don't have the right unless you have a formal permission. anyway if there is a problem or issue related to a call recording in the courts just the official bodies have the right to ask for this recording from the call center to judge on the case that they have. and finally if the purpose was for entertainment this is ethically forbidden. 

Yes. Ofcourse you can. Its your right. No worries.

Most probably IVR said for training and quality purposes and this for following up and assuring the quality of the service they are providing however u can record the call as well but there is no need because if you need simply to make a complaint or ask to prove ur rights the company it self responsible to provide this call in case it was asked by Judges or legal authority

for sure you can record the call. and you can use it legally if necessary after taking permission from the police if required.

as long as there is no jurisdiction in that country prohibiting private recording of calls, it is okay.