What you are describing looks like a bad connection somewhere.
The terminals on the battery may be clean, but they are many other places where there could be a bad contact. Make sure that the connection between the leads and the clamps that go on the terminals is clean and corrosion free, and that the ground to the chassis is also good. Same thing on the wires leading to the starter. Unfasten the connectors, clean and inspect both parts, use a dielectric grease all over the connector, then make sure that you reconnect correctly. Every connection that you inspect like this will be one connection that you'll never doubt again.
If it's all good, then have the battery and alternator tested. Even if the battery is new, it could be bad, for example if the charging system is not up to the task of keeping it fully charged.
The terminals on the battery may be clean, but they are many other places where there could be a bad contact. Make sure that the connection between the leads and the clamps that go on the terminals is clean and corrosion free, and that the ground to the chassis is also good. Same thing on the wires leading to the starter. Unfasten the connectors, clean and inspect both parts, use a dielectric grease all over the connector, then make sure that you reconnect correctly. Every connection that you inspect like this will be one connection that you'll never doubt again.
If it's all good, then have the battery and alternator tested. Even if the battery is new, it could be bad, for example if the charging system is not up to the task of keeping it fully charged.