I would like to expand on Latinumbar's comments regarding space combat & correct those regarding ground combat.
To clear Borg shield neutralizer use Hazard Emitters. Given you you'll get hit with that debuff multiple time, having 2 HE will help a lot.
To clear the Borg tractor beam use Polarize Hull or Attack Pattern Omega. PH will give you only two debuffs - tractor resist & damage resist. I suggest using APO because it provides six buffs - tractor resist, damage resist, DPS boost, flight speed boost, turn rate boost, & defense strength boost. Again have 2 copies of which ever skill you decide to use.
Let's discuss ground combat now. There is a weapon that is the deadliest against the Borg - the Synchronic Proton Distortion Prototype Assault Rifle or affectionately known in the STO community as the "Ghostbuster gun". You get it as the reward for completing the Spin the Wheel mission. Here's the reasons why it is the most effect weapon against the Borg.
As you know, Borg adapt to a specific damage type in 5 shots. The Ghostbuster gun is the only weapon that fires two damage types - the primary is antiproton & the secondary is proton. That means you can fire the primary 5X & the secondary 5X before you have to remodulate. On a team of 5, that's enough DPS to destroy any Borg Patrol, but it gets better.
The secondary attack fires a high density proton beam at the first target & chains up to 3 more targets at the same time. That's an exploit attack against 4 targets. If a team of 5 fires the secondary weapon at one Borg, the entire Borg Patrol will die from that one shot. If any Borg do survive (you took aggro from multiple patrols or your team is firing at multiple targets), spam the primary weapon and clean up what's left.
The Ghostbuster guns does so much damage, you'll only have to remodulate once AFTER your team is finished killing an entire Borg patrol. To do maximum damage you do need to follow a basic strategy. The "chain effect" only works as long as the target you're firing at is alive. Once that target dies, the chain stops. To maximize your damage, fire at the toughest (most HP) Borg first. Your firing order should be Elite, Heavy, Tactical, Medical, & lastly Drones.
I've confirmed the Ghostbuster gun's effectiveness against the Borg. On a team of 5, only equipped with Ghostbuster guns (no secondary weapon), we completed Infected in 20 minutes, the Cure in 30 minutes & Khitomer Accord in 1 hour. We did use other weapons against the Bosses but those battles probably represent less than 1% of the entire ground mission. We were even able to complete a 2 man team Cure run as well using only Ghostbuster guns.
Do the Borg have weakness to any weapon types on the ground or in space? How should I equip my captn and ship to fight the Borg?
You can use anything against the Borg in space, although like everything else Escorts are the most efficient way of destroying them.
For Ground then anything with high damage single shot weapons are very effective; Sniper Rifles, Compression Pistols and Blast Assaults (I don't like the latter personally). If you're a Tac however then you can make good use of pulsewaves, my Tac regularly one shots entire patrols of Borg when fully buffed and because it's all one big hit on multiple targets you'll get the usual 4 or 5 shots off before having to remodulate.
As for equipment, the new STFs (due with F2P) will drop salvage with which you can claim [Borg] weapons that have a chance for bonus damage against the Borg, you'll also be able to requisition ship and ground sets. For now though the Mk X or mk XI weapons that are readily available in game are more than good enough for STFs.
Depending on your ship, you might not have the room for two HE or APO, as a cruiser captain, I rely heavily on hull tanking. You can clear one shield drain with HE, but especially in Red Alerts you'll be spammed with them, so it's pointless to waste your time caring about the shields when your hull is strong enough.
That means you should have two Engineering Teams and good armor. I recommend ablative armor, as it covers most types of damage, perhaps neutronium wouldn't be too bad, too, to have additional protection against torpedos.