Can I go for both Tichu types in one round?
No, you can only call one type of Tichu per hand.

Honestly, I spent an inordinate amount of time looking at different rules (here, here, here, and here). Ultimately, none of these explicitly state whether a player is allowed to call both types of Tichu, but all of them imply it. I finally found one source with this wording after Grand Tichu:


  A single player may not call both a "Grand Tichu" and a standard "Tichu".


This seems to explicitly call out what the other rule sets imply: after calling a Grand Tichu, you cannot call a small Tichu, and vice versa (assuming you're calling a small Tichu before you see nine cards). The website these rules come from also has some strategy documents that appear to support it's claim that two Tichus cannot be called by a single player.



On a separate note, part of the reason this might not be explicitly called out in many rule sets is because they are translations. I don't read German, but someone might be able to verify that the rules as originally written indicate that only one Tichu can be called, and translations failed to pick up on this.