The most common reason a strong appellate argument is unavailable is that nobody made it below. Appellate courts review a closed record, and with narrow exceptions will not consider arguments raised for the first time on appeal.
Object, specifically. A general objection preserves little. State the ground. If the ruling is against you, you have preserved that ground and generally not others.
Make an offer of proof. When evidence is excluded, the record must show what it would have been, or the appellate court cannot assess whether exclusion mattered.
Get rulings. An objection the court never rules on may not be preserved. Ask for the ruling and make sure it is on the record.
Renew where required. Some motions must be renewed at the close of evidence to preserve the point; a motion made and never renewed can be treated as abandoned.
Object to instructions before they are given, with the substance of the correct instruction stated.
Watch the record itself. Sidebars off the record, exhibits not formally admitted, and proceedings the reporter did not transcribe are all gaps that cannot be filled later.
The practical recommendation is to have someone thinking about the appellate record while the record is still open. Preservation decisions are irreversible once the trial ends, and no brief written afterwards can repair them.