The 3rd level ability allows a Beast Master Ranger to "Choose a beast that is no larger than medium and has a challenge rating of 1/4 or lower" and "If the beast dies, you can obtain another one ..."
"Another one" means the same type of beast as the first i.e. no larger than Medium with CR 1/4 or less.
As an aside: the Beast Master Ranger is a relatively low-power class option; particularly if the ranger has no beast companion! Your DM should ensure that you can replace the beast as quickly and as easily as possible - right now you are about as much use as a Wizard without spellbooks, a blind fighter or an amputee rogue.
It gets the benefits immediately
You get the Companion's Bond feature at 3rd level, at the same time as you gain the Animal Companion feature.
The Companion's Bond feature says:
In addition to the areas where it
normally uses its proficiency bonus, an animal
companion also adds its proficiency bonus to its
AC and to its damage rolls.
This applies from 3rd level and onwards, so it gets a +2 to its attack rolls, ability and skill checks, saves, AC and damage, and loses Multiattack if it had it.
Once you reach 4th level, and every level beyond that, it gets an extra hit die:
For each level you gain after 3rd, your animal
companion gains an additional hit die and
increases its hit points accordingly.
This means that your giant weasel's stats look like this:
Giant Weasel
Medium beast, unaligned
Armor Class 15 (13 +2 proficiency)
Hit Points 9 (2d8)
Speed 40 ft.
Ability Scores
(+2 proficiency to all saves)
STR 11 (+0)
DEX 16 (+3)
CON 10 (+0)
INT 4 (-3)
WIS 12 (+1)
CHA 5 (-3)
Skills Perception +5, Stealth +7
Senses darkvision 60 ft., passive Perception 13
Languages —
Challenge 1/8 (25 XP)
Keen Hearing and Smell: The weasel has advantage on Wisdom (Perception) checks that rely on hearing or smell.
Actions
Bite: Melee Weapon Attack: +5 to hit (+3 +2 proficiency), reach 5 ft., one target. Hit: 7 (1d4 + 3 + 2 proficiency) piercing damage.
Best Answer
No.
But its AC, attack rolls and damage scale with the ranger's proficiency bonus, and its HPs with ranger's level.
PHB p.93