What is one to learn after Birchas Hatorah?
The custom is to recite the verses of Birchas Kohanim [and the Mishna of Eilu Dvarim] immediately after reciting Birchas Hatorah, and through doing so one fulfills his obligation of learning Torah immediately after the blessing. Nevertheless from the letter of the law one may learn any part of Torah, whether Scripture, Mishna, Talmud, or Midrash, as all the above is considered Torah and was given to Moses on Sinai. One is not to fulfill his obligation of learning Torah after Birchas Hatorah by writing words of Torah after the blessing unless he verbalizes the words that he is writing.
 The reason for reciting specifically Birchas Kohanim: As the Sages desired that one read at least three versus from the Torah just as is done when reading from the Torah scroll. [Divreiy Chamudos Brachos 1/75] Likewise it is because it contains words of blessing. [M”B 47/20] And is known by heart and is permitted to be recited by heart. [Eretz Tzevi 1/27]
 Siddur, not explicitly mentioned in Shulchan Aruch Admur, however see 47/3; brought in M”B 47/21. The saying of this Mishnah after Birchas Kohanim is brought in Rambam Nussach Hatefilah as a custom of all Israel. M”B 47/21 states it is recited in order to fulfill the reading of Mikra Mishna [Eilu Dvraim Sheiyn Lahem Shiur] and Gemara [Eilu Dvarim Sheadam Ochel]. See Piskeiy Teshuvos 47/12 that the reason this Mishnah was chosen to be said is because it is easy to understand, and emphasizes the greatness of Mitzvos of man to man and the importance of Talmud Torah. [Likkutei Mahrich in name of Rav Zusha of Anipoli]
May one say Birchas Kohanim after Birchas Hatorah if it is before Alos? Being that these verses are being said for the sake of learning (and not for the sake of Nesias Kapayim) it may therefore be said even prior to daybreak even though one does not do Nesias Kapayim at night. [ibid] This follows the ruling of the M”A 47/8 and is unlike the ruling of the Rashal.
There is an opinion [Tashbatz; Rav David Abudarham; Michaber 47/3; Aruch Hashulchan 47/10] which says that one must say a blessing prior to writing words of Torah even if he does not verbalize the words that he is writing. Nevertheless this only applies if one understands and is paying attention to the words that he is writing. If however one does not focus on the words he is writing, as is sometimes the case when one is copying something over into writing, then one may write words of Torah before saying the blessing. Similarly if one is writing a letter to a friend he may write verses for the sake of formality of the letter before saying the blessing, as he has no intention to do so for the sake of learning Torah. Practically one should not rely on this opinion to recite Birchas Hatorah prior to writing Torah alone, as this is merely like one is thinking Torah [which one may do before saying the blessing]. Rather it is proper for all those that write Torah after Birchas Hatorah to verbalize a few of the words that he is writing in order to avoid a blessing in vain. This however only applies if he did not learn anything in-between the blessing and the writing. However in a case that one said the verses of Birchas Kohanim or the Mishna that is accustomed to be said after reciting Birchas Hatorah, as is written in the Siddur, then he does not need to verbalize the words which he is writing. [ibid]