BglII A'GATCT/TCTAG,A and BamHI G'GATCC/CCTA,G are compatible so you cut your vector with BamHI and EcoRI and ligate with/insert your fragment BglII-EcoRI
in a microtube, you digest your pUC19 vector with both enzymes (Bam HI and Eco RI) if their incubation media are compatible, separately with the less salt medium first if the incubation media are too different ; you may even dephosphorylate the vector after digestion to optimise the cloning of your insert ; after ethanol precipitation and washing you resuspend in water ; similarly, the insert is digested with both enzymes (Bgl II and Eco RI) if their incubation media are compatible, separately... ; the insert (I don't know it comes from...) may be eluted from an agarose gel to select the good length, etc. ; ligate them (vector + insert) following standard protocol and select white colonies (lac-) with as control the vector only (which after dephosphorylation would give very few blue transformants).