I. What is the שיעור of a תרועה?
The Mishna in Rosh Hashana(33b) states that the שיעור of a תרועה is שלש יבבות. Rashi understand this to mean three very short sounds. Tosafos disagrees with Rashi and holds that each יבבה is 3 short sounds meaning that a תרועה is 9 short sounds.
II. What is the שיעור of a תקיעה?
The Gemara states that שיעור תקיעה כתרועה. The simple understanding is that a תקיעה has to be the same length as a תרועה. Therefore the מחלוקת Rashi and Tosafos about תרועה applies to תקיעה as well. According to Rashi a תקיעה is very short, the length of 3 short sounds. According to Tosafos it is 3 times as long the length of 9 small sounds.
The Rambam has a unique שיטה. The Rambam understands that when the Gemara said that שיעור תקיעה כתרועה it meant both the תקיעות that relate to a תרועה as every תקיעה has a תרועה before and a תרועה after. Therefore the שיעור of a תקיעה is half a תרועה.
III. שברים
Tosafos points out that according to Rashi each שבר of a שברים has to be the length of 2 and only 2 short sounds. Tosafos has the following simple calculation.
1. Each sound of a תרועה is 1, therefore each שבר needs to be longer otherwise it will be a תרועה and not a שברים
2. A תקיעה is the length of 3 sounds so therefore each שבר must be shorter then that otherwise it will be considered a תקיעה and not a שברים
3. Therefore each שבר must be 2.
Tosafos has less of a problem because according to them the minimum תקיעה is 9 so a שברים just needs to be less then 9.
IV. שברים תרועה
The Rishonim have a major מחלוקת how you blow it, בנשימה אחת or שתי נשימות?
The Machlokes is based on a Gemara in Succah(53b). The Gemara there discusses the תקיעות in the מקדש. R' Yehuda holds that תקיעה תרועה תקיעה is 1 מצוה and therefore אין בין תקיעה לתרועה ולא כלום.
The Ritva, Ramban and others learn as follows. They read the Gemara literally that according to R' Yehuda there is absolutely no הפסק between the 3 קולות because they are 1 mitzva. Therefore they say even though we don't paskin like R' Yehuda, his din applies to שברים תרועה. Everyone agrees that שברים תרועה is 1 mitzva and therefore you cannot be מפסיק and שברים תרועה is בנשימה אחת.
Rashi and Tosafos in Succah learn that אין בין תקיעה לתרועה ולא כלום means you can't be מפסיק more then a נשימה. However, since these are separate קולות you are מפסיק a נשימה between them, which means that they hold שברים תרועה is בשתי נשימות.
There also is a מחלוקת what does בנשימה אחת and שתי נשימות mean. The Mishna Berura writes that נשימה אחת is if you were מפסיק less then a נשימה. The חזו"א disagrees and says that נשימה אחת is no הפסק whatsoever (and this is what the Rishonim seem to say). By שתי נשימות you have a similar dispute. The חזו"א holds that it means a הפסק but not an actual breath and may others (Shulchan Aruch Harav, מקראי קדש, etc.) hold you should specifically take a breath.
What sound is a שברים? From Tosafos it is clear that a שברים is a very short תקיעה as according to Tosafos if you make a שבר too long it becomes a תקיעה. The Ramban however, writes that a שברים should be a broken sound as well. Many have a minhag to blow both types of שברים.