as long as you are illegal, you don't have a chance to do Ausbildung at all. If some firm or school permits you to start Ausbildung, they make a criminal case and risk a penalty.
Besides to make Ausbildung in Germany you need at least a B1 level of German, mostly B2. It depends on the subject.
The new law regulates a so called Ausbildungsduldung, if you mean this. Only people who started the Ausbildung during their asylum procedure, can get Ausbildungsduldung immediately (provided, they fulfil all conditions like a passport or clear identity, no criminal cases etc.). If you only have an offer for Ausbildung and want to apply for Ausbildungsduldung, you must first have a normal (risky) Duldung for at least 3 months, then you can make such an application. The problem is that, if you go out of illegality, you must first have a permission for any occupation inclusive Ausbildung (also a kind of occupation). Illegality is a criminal case and if you get punishment for that, it can be a problem with a permission for occupation (work, Ausbildung). And, naturally, there must be reasons against deportation, other than not having proper documents (because not providing the authorities with identity documents means that you won't have a permission for occupation).
If you make an asylum application, you normally must live in a camp for many months and in this time you have a long ban on occupation anyway.
It is a very general information which is perhaps very disappointing. Each case is special, you need a good individual consulting. Please look for a good councelling center in your place. The best thing for you is naturally to take a lawyer because stopping to be illegal can bring a number of special problems.
There are generally 2 ways.