I'm going to go against the general direction of the question and answers here:
You need a pointy stick
Pretty good to have extras with it
- Long stick
- The point being made well.
Behold, a blood mage (artistic take):
When it comes to weapons, the spear is pretty good:
- It is very easy to make (literally a pointy stick)
- It is easy to use with little training
- It can cause quite a lot of injuries
- It can be used more tactically to trip opponents
- Can be very light, so easy to wield and just carry
- Most importantly, it keeps you away from harm
Since a blood mage's fight would basically be:
draw blood -> use blood -> repeat but with blood
Then it stands to reason that the mage would want to avoid damage as much as possible. A spear is great at that. A shield can be added for extra protection:
(being made of plastic helps with avoiding injuries but it's not a requirement for the mage)
At any rate, the aim is
- Stay safe.
- Stab the opponent.
- Follow up with blood manipulation.
You can take a stab during the blood magic casting, if possible but not requirement.
Here is a video (14 min) of reenactors trying spears versus swords in different ways (and the longer version (28 minutes) showing more of the actual matches) and spear fares pretty well.
Now a spear is not the ultimate weapon but it's still very good. Most other weapons require a lot more training to get to use well. Especially well enough to fight other people. I posit that a mage who likely hasn't that much of combat experience. At least, probably not as much as somebody who comes at them swinging a sword does. So, a spear has many advantages in being simple, effective, and putting you away from harm's reach.
There are some drawbacks.
Heavier armour would pose a problem. My suggestion is to run. Yeah, an amazing tactic, right? But why not run? Even with a sword or another weapon, you're not going to be a lot more effective against an armoured opponent. The weapons and techniques good against armour put you at a lot more of a risk of injury and death. And weapons like clubs and maces that are decent against armour don't even allow you to use the biggest advantage - blood magic, as they'd cause more internal bleeding than what external, so not usable.
If you really have to face an armoured opponent, then...well, actually a long pointy stick is not even that bad, honestly. You can use a polearm of some sort
This is a poleaxe. It's also informally known as a "can opener". To be blunt (unlike it) this is because it's quite effective used against heavily armoured opponents. You can use the axe-like blade as a hook to drop a heavily armoured opponent to the ground. Now, they are not quite helpless but certainly a lot more vulnerable than before.
Also worth remembering that you can still have other weapons with the spear. If your mages are really amazing at fighting, then they can still use a sword, or anything else. It's still beneficial to stay away from your opponents at first but you should be able to drop the spear and draw a different weapon, if needed.
Just as a suggestion, a rather gruesome weapon a blood mage might want to use is some version of the katar
It's a type of punching dagger. You grab the handle and the blade is now an extension of your forearm. You can strike with this and it's even rather good at piercing armour. It's also very messy and will leave the victim profusely bleeding. You can get wavy bladed ones that cause even more injuries or...you can get a scissor katar. It generally looks the same as above when folded but you can enable a mechanism that "opens" it and two blades spring to the sides. It looks a bit like this:
It becomes a way messier weapon. Pulling it out after opening the blades will cause even more damage and very rapid blood loss. Seems fitting for a mage who could use a lot of blood. I imagine the blood mage will use the katar to supply himself with access to a lot of blood to use against others. So it could be a good weapon to start an engagement with but it requires very close proximity. Luckily, it's easy to conceal under a sleeve, so if engaging multiple opponents and the mage has the advantage of surprise, this can work very well - stab one, use the blood of the first victim against the rest.
If the mages are supposed to be really good combatants and you don't want them to use spears, then I can suggest the butterfly swords:
This is a Wing Chun weapon. Two swords supposed to be used as one - some even fitted a single scabbard together (as pictured here) others came with separate scabbards. At any rate, the main idea of Wing Chun is to deflect blows and gradually wear out an opponent through small but repeated counter-attacks. The two swords fit exactly this purpose. The way you're supposed to use them is to swing both blades together in a single direction. You deflect a blow coming to you and use the small opening to nick the opponent's arm. They swing again, you deflect and you nick again. And again. With each successive blow on your part, the opponent feels a little more pain and the cuts make it so their blows are easier to deflect. Which, in turn, gives you better counter-attack opportunities, so you can "move forward" with your cuts.
Having two blades means that you can deflect from either side and you'd always have a free sword to do a quick slash. The catch behind the blade (the bit that sticks out) can be utilised to trap opponent's weapon between it and the backside of your blade. You could just use it to lock the weapon for an extra moment with one hand and attack with the other, or use both swords for a better leverage to disarm an opponent.
So, butterfly swords do take some training to use. However, they are rather effective at defence and drawing blood repeatedly. That is the way you're supposed to use them, after all. Using the butterfly swords also looks really cool (2 min video)