$ awk -F':' '{$0 = (NF>1 ? ">"$2 : $0)} 1' file
>TRINITY_DN10034_c0_g1_i1
CCAGCGCCACGGAGGGCGAAGGCGAAGGCGGTGGTTTTGAACGATGAGGTGCCGCCGGTG
>TRINITY_DN10322_c0_g1_i2
GATCCGCCTTCTATAACTGGAAAAGAAGAGGATGAGGTGCCGCCGATGAGTCTACCTTCT
$ awk -F':' '{sub(/>.*/,">"$2)}1' file
>TRINITY_DN10034_c0_g1_i1
CCAGCGCCACGGAGGGCGAAGGCGAAGGCGGTGGTTTTGAACGATGAGGTGCCGCCGGTG
>TRINITY_DN10322_c0_g1_i2
GATCCGCCTTCTATAACTGGAAAAGAAGAGGATGAGGTGCCGCCGATGAGTCTACCTTCT