Hi, as an actual owner of a Taurus TCP 738, I guess I can be of help.
First, the gun needs to be broken in. Shoot a few hundred target rounds (FMJ) through it. When it reliably cycles those, shoot at least 25 self defense rounds (JHP) trough it to make sure it cycles those. .380 Auto and .380 ACP are the same thing.
Hand cycling ammo through the Taurus TCP 738 requires to mimic the sharp rearward movement of the slide. This has to do with how the ejector works. Pulling the slide back too slowly will result in the round getting extracted and stuck, instead of being properly ejected. This can happen with some self defense (JHP) ammo, such as Hornady Critical Defense.
The magazine springs can be initially quite stiff, which can result in the top round pushing too much against the cocking rail. This can slow down the slide through friction and can cause FTFs and FTEs. Leave the magazines loaded to give them some static break-in. The magazine springs will wear in after being loaded for a month.
I noticed a rather sticky surface coating on all my new Taurus TCP magazines. I disassembled them and gave them a good cleaning with Break Free CLP. I even polished the inside slightly with metal polish. This assures that the follower has less friction.
Further tuning could include some surface polishing and edge stoning. There are a few instructions at
TCP 738 reliability enhancements and
Taurus TCP-738 .380 random jamming fix.. (pics) - Page 2. It includes polishing the feed ramp, the cocking rail, the hammer, the breech face, and the outside of the barrel. It also includes stoning some of the edges on the slide, the extractor, and the loaded chamber indicator. This may not be necessary for your gun and would require some expertise. Detail stripping the TCP has its challenges.
I did add a Crimson Trace CTC Laserguard,
Galloway Precision :: Taurus® Performance :: TCP Performance Parts :: +1 Magazine Extension for Taurus TCP to my magazines (now holds 7+1), and a Pachmayr Handgun Grip. The grip extension really helps to get a good grip when drawing and helps to avoid limp wristing. The Handgun Grip allows for a more firm and secure grip while shooting. For jogging I am using a Rangemaxx Inside the Pants Handgun Holster designed for small 380 pistols with a Laserguard. It is $14 at Bass Pro. I removed the Velcro retention strap and the belt loop on the side that is facing my body. For hot weather I use a CrossBreed MiniTuck holster. The factory sights are pretty useless, so I added
Nitesiters Handgun Night Sights. I also recently installed the
Galloway Precision :: Taurus® Performance :: TCP Performance Parts :: Sweet Spot Trigger for Taurus TCP. Alabama Holster Company makes good
Link Removed for the TCP and other pocket guns.
As for self defense ammo, take a look at
Final Results of the .380 ACP Ammo Quest | Shooting The Bull.