You do not gain any Ability Score increases off the bat. Instead, at 5th Level, you can pick to learn either "Growing Body" or "Lengthening Body"

Nagas slithering speed is normally at 30 ft.
However, once they brought their heavy bodies in motion, they may move up to 45 ft. on their next turn if they have moved on the previous turn.
Additionally, a Naga cannot stop on the spot after taking advantage of this effect. On their next turn, they need to move for at least 15 ft. in a believable direction until they can halt. If this is not possible without hitting an obstacle, they take 1d4 bludgeoning damage per 5 ft. of the remaining 15 ft. "slow down" movement. The obstacle takes the same amount of damage. Depending on the obstacle, they may come to an immediate halt or push it with them. This is at the DMs discretion.

Due to your unique body shape, difficult terrrain does not affect your movement and shallow water does not slow you down.

Due to their nature and anatomy, Nagas posess the ability to talk to wild snakes and snake-like creatures and understand them as well. 

Due to their lengthy body, Nagas are classified as large creatures and take a 10x10 ft. space on the grid. Unless specified otherwise, when remaining still, Nagas will curl their tail around themselves beneath them. If specified otherwise, the DM and player can decide what squares they occupy on the grid.

Due to the unique shape of your snake body, most armours for humanoids wont fit you completely. Any armour that is full bodied receive a reduction in AC until you have either modified the armour yourself or paid a smith to. The reduction goes as followes: Light Armour: -1AC, Medium Armour: -2AC, Heavy Armour: -3AC.

Nagas have a pair of fangs instead of traditional canine teeth with wich they can deliver a bite attack that deals 2d4 piercing damage. At will, they can also poison their target with a successfull bite attack. This deals 2d8 poison and can poison the target. The saving throw is CON and needs to equal or exceed 8 + the Nagas CON Modifier.
The poison aspect of the bite attack can only be used once per short rest.

Nagas are also immune to poison themselves.

Nagas can also extract their own poison instead and store it in a flask. This also requires a short rest.

At Level 3, a Naga gains a better understanding of their body. They can take advantage of their "Heavy Body" Trait and use it more effectively. They only take half the damage they normally would.

After reaching full maturity in their late teens (like most other humanoids), Nagas stop ageing in a traditional sense. Instead, their body keeps growing permanetly until they reach roughly 30 ft length at ~300 years of age. After that, the growth progress is extremely minor as it slowes down exponentially.

The benefits result in +1 STR and +5 ft. Movement speed (of any kind of movement) at 5th Level, 11th Level and 16th Level.

After reaching full maturity in their late teens (like most other humanoids), Nagas stop ageing in a traditional sense. Instead, their body keeps growing permanetly until they reach roughly 30 ft length at ~300 years of age. After that, the growth progress is extremely minor as it slowes down exponentially.

The benefits result in +1 DEX and +5 ft. of movement (applies to all movement) at 5th Level, 11th Level and 16th Level.

At 5th Level, a Naga learns to utilise its bodies strength much better. It gains the ability to ensnare and constrict a target.

To ensnare a target, the target must fail a DEX save of 8 + the Dex Modifier of the Naga + its Proficiency Bonus. At 12th Level, you gain Expertise with your ensnaring move and add twice your proficiency modifier instead. 

Ensnared - If a target is successfully ensnared, it must make a Strength Save with a DC of 8 + the Strength Modifier of the Naga + its Proficiency Bonus. At 12th Level, you gain Expertise with your constricting move and add twice your proficiency modifier instead. If the target fails, it becomes constricted.

Constricted - A constricted target cannot move its arms and attack with any weapon that it has in its hands, neither can it cast a spell that requires a somatic component. Its movement of both (Naga and its target) is reduced to 0. For the next 3 turns, it has to repeat the strength saving throw. to escape the constriction. While constricted, the Naga may attack the target without an attack roll. However, all attacks on the Naga are made with advantage. Furthermore, of the AC if the target is higher than the Nagas AC, all attacks that are equal or higher than the Nagas AC, but lower than the targets AC hit the Naga instead.
After 3 failed strength saves while constricted, the target will start to choke. It looses the ability to cast spells that require a verbal component and continues its strength saves at disadvantage for an amount of turns, equal to its constitution modifier. Should it fail all these, the target goes unconscious or dies (players choise). If the target is released in this state, it remains unconcious for 1d8 hours. Any potential injuries, like broken ribs can be added as flavour but should not impact the targets health significantly, or bring any long term consequences.

NOTE: If the Naga, at any time, takes either a critical hit, or more than 50% of its maximum HP in a single hit or more than 75% of its remaining HP as damage, it must let go of the target or succeed a Strength save equal or higher than the taken damage.

Recommended for Nagas with slender bodyshapes.

At 5th Level, you can learn the Tail Sweep.

Your tail sweep is a 270° circular attack with your tail. Its range depends on your total length minus 8 ft. Your enemy needs to succeed a DEX saving throw of 8 + your Dex Modifier + your proficiency or fall prone. Additionally, they take 1d8 bludgeoning damage (at Level 5, 1d12 at Level 11 and 1d20 at Level 17).

Additionally, after performing this attack, you immediately gain your full movement of 45 ft.

Furthermore, you get +1 DEX and 5 ft of movement (applies to all movement) at 5th Level, 11th Level and 16th Level.

At 7th Level, a Naga can raffine their extracted poison and create a more leathal version that deals 4d8 poison damage instead.

They als may add their proficiency modifier to the Saving throw to resist the poison.

At 17th Level, a Cursed Naga will undergo a 3-Day period of many changes once it feels ready and sufficiently prepared.

Day 1: The Naga will grow in size, up to an additional 15 ft. in length.
Day 2: The Naga will finish growing and they become a Hermaphrodite (if not already by birth). [Hermaphrodite means that they posess male, as well as female, functional genitalia]
Day 3: The Naga will learn to get used to its new body quickly. It learns either "Growing Body" or "Lengthening Body" (depending on what the Naga has not learned yet) and gain its special attack trait as well as the Ability Scores coming with the traits.