Question: I cannot keep up with the pace of davening at our minyan. Is it okay to start the amida early so that I can be finished in time for kedusha?
Answer: The Gemara (Berachos 8a) teaches the importance of davening with the tzibbur whenever possible. Rambam (Tefilla 8:1) codifies this as a halacha. The Shulchan Aruch (OC 90:10) adds that one is not allowed to begin their davening earlier than the tzibbur unless it is getting really late due to adding in various piyuttim or for some other reason.
The Chayei Adam (19:1) writes that tefilla betzibbur specifically applies to the amida. One should be particular to come on time to shul so that they can maintain the pace of davening.
The Pri Megadim (Eshel Avraham 109:2) writes that one only fulfils their obligation of tefilla betzibbur when they begin the amida at the same time with the tzibbur. Nonetheless, R’ Moshe Feinstein (Igros Moshe OC 3:4) writes that as everybody davens at a different pace, the tzibbur will not necessarily be reciting the same berachos as each other simultaneously. Therefore, it does not matter if one begins a little later.
The Ben Ish Chai (Ki Sissa 1:6) writes that one who davens slower than the tzibbur and so does not listen to the chazaras hashatz, is considered to have participated in chazaras hashatz, too. Accordingly, they should start their amida along with everyone else (See Aruch Hashulchan OC 109:5).
Nonetheless, R’ Ovadia Yosef (Yabia Omer 2:7) and R’ Moshe Stern (Baer Moshe 4:9) write that one who davens slower than the tzibbur can start their amida earlier. Doing so still constitutes tefilla betzibbur. The Bach (OC 236) writes that the gabbai can start davening amida early on Rosh Chodesh so that he can remind people loudly to recite yaaleh veyavo. Clearly, he maintains that doing so still constitutes tefilla betzibbur.
One reason to start at the same time is out of respect for the tzibbur. When it is apparent that one starts early so that they can keep up, they are not disrespecting the tzibbur. Therefore, one who davens slower than the tzibbur may begin their amida earlier to be able to recite kedusha. One may not do so in mincha, however, as it is more important to respond to kaddish than to recite kedusha.
In conclusion, one who davens slower than the tzibbur may begin their amida before the tzibbur. They should not do so in mincha (or maariv) as doing so will mean forfeiting kaddish.