The beginning of Tazria speaks of how a woman is tamei for seven days with a boy. This is then followed by the repeat of the commandment for bris milah on the eighth day. The meforshim state that this allows the woman to attend the bris and that this now causes the child to be part of klal Yisroel. The Milah is then followed by the remainder of her tum'ah until she brings a sacrifice and the halacha for having given birth to a girl. After that, we have the halachos of Tzora'as (usually tranlated inaccurately as 'leprosy', see Rav Hirsch for example for this discussion). Many meforshim ask why are the 'extra' halachos of bris milah (see sifra and maseches Shabbos 132a) given here rather than continuing the flow of tuma'a from the woman being tamei to the laws of Tzora'as. Why does Bris Mila get inserted.
Rav Samson Rafael Hirsch has stated that the numbers 6, 7, and 8 connect with Maasei Bereishis [creation] to show the way a person exists. The number 6 is the creation of the natural world. It is the set up of the laws and instincts that allow the physical world to continue and the living beings in it to exist. Shabbos, as the number 7, symbolizes the completion of the natural world and the continuation of nature without new 'explicit' creations. The number 8 therefore, symbolizes "L'ma'alah min hatevah" [above or outside of nature]. That is the beginning of a new cycle, showing a raising of human status so that Man, unlike the rest of nature can change. Man can become 'greater than the mala'chim or less than the animals'. This is hinted at in the first Rashi of Parshas Tazria which states that Rav Simlai explained that this is connected to the order of creation in which Man was created after all the animals. Just as Man was created after all the animals, the parsha of giving birth comes after the explanation of taharah for the animals. Bris Milah is the next step (L'maaleh min Hateva') of Bnai Yisroel and is required before tzora'as can occur.
We see that we are told to learn various traits from different animals, 'faithfulness from a dove', 'modesty from the cat', 'industry from the ant', ... These traits are hard wired into the animals and are not something that they "choose" to do. However, we can see the traits and learn from them.
We can also see a reason for putting Milah in the Torah before Tzora'as because tzora'as is a completely miraculous occurrence which has not connection with the natural world. We see this because it only occurs among Bnai Yisroel. Non-Jews do not (normally - except for Naaman with Elisha) get this 'disease' and it is not treated according to the normal laws of epidemiology. It only existed during the time that spiritual matters had obvious physical effects. As a result, it is completely l'ma'aleh min hateva. A person is not quarantined for health related purposes, as was done with measles or tuberculosis. A person can only be quarantined upon declaration of the kohen and the kohen is forbidden to declare a person tamei during the chagim or during the seven days after his wedding (when health reasons would make it more necessary).
We can compare this to the halachos of tum'ah as given for people and animals. Only someone who is able to become kadosh is able to become tamei. Tum'as neveilah [ritual impurity caused by an improperly killed animal] applies only to kosher animals (that can be slaughtered and eaten). Nonkosher species do not have this kind of tum'ah apply to them. Vegetables cannot become tamei until after they have been washed and 'made ready' (machshir le'tum'ah). Similarly, a person who can become 'greater than the angels' is also capable of becoming 'lower than the animals'. Thus, in order to be subject to the tum'ah of tzora'as, a person must become elevated to the next level of kedusha (through bris milah for a boy). This can explain why we must have the halachos of milah at this point of the Torah.