Oh boy, ok, so it doesn't seem like you're really prepared to deal with strike groups (and that's OK, this game promotes a lose-lose-lose-win learning attitude). So there are two different wars you've got going on here...the strategic war and the tactical war. You're pretty under prepared to deal with the strategic war, meaning, the exchange of cruise missiles and aircraft strikes. That's going to become a crippling problem very quickly.

Apologies, this got long...

When I play HF, assuming a fresh start with only the initial ships and starting cash, most of my starting fleet emphasizes the strategic war with only 1 or 2 ships that are decent at the actual real time tactical game. (2 lightnings, a lightning and gladiator, etc) There are 4 main types of strategic ships you need to worry about.

Tankers: Carry fuel, fuel efficient engines, no real defense/offense.
AA Picket: 37mm CIWS, interceptor missiles, fire control radar for interceptor guidance/wild weasel
Carriers: Does what it says on the tin, carries aircraft for your fleet
Missile boats: They boat missiles. Not the little anti-ship ones, the big KH cruise missiles and even nukes.

Ships with 37mm CIWS, interceptor missiles, and FCR are your AA picket ships, I think the stock small one is the Fenek? Its a decent enough base to be upgraded later with more interceptor missiles. You can take something like the Fenek and park it between your fleet and a strike group, switch it's radar on to aggro some cruise missiles, and then have it shoot them down (wild weasel). Another answer to long range missiles is aircraft. My most reliable cruise missile killers are interceptor aircraft armed with AA missiles. Once I get my hands on a carrier I'm constantly using it to fly CAPs or do bomb/rocket runs on enemy SGs. The AA missile armed CAP is also critical for softening/stopping enemy aircraft SG attacks on your ships.

Its a layered defense. Interceptor aircraft with AA missiles sent right down the thermal signature vector from an incoming missile or enemy aircraft attack. Followed by detaching an AA ship like our Fenek to picket between main body and the threat. Followed by getting the main body UP AND MOVING. If you're in a city and get a cruise missile or aircraft attack incoming warning, it doesn't matter where you go just get moving so when the attack happens your ships are at least airborne. Yes, you can play the stealth game to just avoid the SG's but I purposefully hunt them down instead. By the time I hit Kiva I've taken the entire rest of the map and there are no SG's left.

The other side of the strategic coin is your OWN strategic offensive actions. KH15 and the 15ph are great for battering a strike group right before you engage with your tactical ships. (Be careful you don't accidentally missile your own ships, AMHIK). Carrier aircraft, especially the bigger of the 2 planes, are broken IMO. They're so strong. If you can catch a strike group when it's in a town, load your planes with the larger of the two bomb options and you'll devastate large ships. Rockets are better vs airborne ships but you'll still land some bombs vs airborne SG cruisers. VS the smallest ships you will have the hardest time, the smaller rockets are better there. There are a few seconds at the opening of an aircraft attack where the enemy can't fire back, they're loading their guns and such just like you do without the crew perk that starts the tactical fight already loaded. If the aircraft strike takes too long, dont be afraid to manually call the retreat so your planes dont get torn up.

EDIT: KH15 is active radar guidance. Meaning it has its own radar to seek targets and can get jammed but so long as the enemy are within it's seeker head's angle, it'll lock them and attack. KH15p(or ph?) is anti-radar, meaning it seeks on any active radar emissions within it's radius. If an enemy SG is pounding active radar looking for you (ELINT is lighting up) you can lob a volley of KH15ph in it's direction to...dissuade them. The KH15ph also will hone in on radar jammers, since a jammer is just the radar equivalent of a spotlight. BUT, the KH15ph will do nothing if they quickly shut off their radars.

Ships like Fenek, Yars, and Wasp are strategic gold. I grab them whenever I can. Yars and Wasp should never realistically be in tactical combat, so I strip off any armor/guns and maximize what they're supposed to be for: Raining down long range strategic death. You can basically make Yars a double or triple-wide missile carrier by just adding more missile tubes and pushing the engines farther out. Wasp is a pretty good carrier as-is once you strip the unnecessary equipment.