ON ACTING: 'Beginning' a Scene
To begin with, I would start correcting the problem of initial boring-ness by erasing the idea of 'starting a scene' from your mind. Acting is life. And life never just starts. It is a continuum. Granted, at a certain point in our life--a certain point in the life of a character--people (audiences) start paying attention. The curtain rises, or the camera is turned on. But who and what we are-- before the scene and moment the scene 'starts'--is the sum total and ongoing energy of all that has gone before...we are at any moment of our lives the sum total of the bundle of emotion that has propelled us into the scene in the first place.
If we have walked up three flights of stairs, we should probably 'enter' the scene perhaps tired and breathy (unless we are playing a phenomenal athlete); or if our boyfriend has left us three hours before, we may 'begin' the scene sad and fighting not to show our sadness; or if we have raced across town to get to the house, we 'begin' the the scene bursting with the energy and emotion of the cross town travel...if you're in Los Angeles, you're still dazed by insanity...melded with the emotional need and objective which is aimed at the person we are talking with, including the style of manner appropriate to our dealing with the other character(s) in the scene.
Don't 'begin' a scene...just allow the camera or audience to pick you (the charcater) up in the scene.
Point two: perhaps another reason you are more interesting and connected in the second half of the scene is that generally the characters--including yours--are by the second half of the scene, and the script, now talking about the central issue in the scene, and the emotion underlying the scene has arisen to the surface, energizing you (the actor as the character) to participate in the life of the scene.
To wit: adjstment. Don't wait until the second half of the scene (when the taxt forces you to get emotionally involved or to get interested) to get stirred up; prepare that emotional/issue involvement far earlier in you...like before the scene starts! The emotion underlying the whole scene has been (should be) present in you (the character) before the beginning, before the curtain rises or the director says "Action". Just because a person gets angry in the middle of the scene does not mean the anger--the potential to be angry--has not bee swimming around in you UNDER THE SURFACE before, perhaps all day...which makes for the audience focusing in on an interesting character right from the start. Two characters may not kiss until the middle or end of the scene, but the sexual desire and tension has been there from the beginning of the scene. A volcano may erupt at the ends of the scene, but a volcano has been rumbling deep in the earth for some time before. And when the audience--as well as you, the actor/character--feels and senses that rumbling before the so-called 'start' of the scene--all present are interested and interesting when the scene 'starts'.
Silly, arbitrary and helpful Rule: start activating the character's emotion and needs at least two minutes before the official 'start' of a scene, and you will find that that official 'start' of a scene will no longer be as boring and dull and uninteresting to either you or the audience as you were before.