I have caught most of the Pokemon including Mew, but I cannot find the..

You have to trade them from other versions.

Most of the Pokemon that you are missing cannot be found in Pokemon Red.
These are:
Weedle, Kakuna, Beedrill, Ekans, Arbok, Koffing, Weezing, Jynx, Meowth, Persian, Electrabuzz, Magmar Raichu (you can't evolve the Pikachu). You'll have to trade these, or, if you haven't beat the Gambler that you use for the Mew cheat, you can use different trainers, like the ones along Cycling Road can give you Electabuzz, Magmar, and Hitmonchan that I know of.
Psyduck can be found on the way to Vermillion from Saffron. You have to be SURFing to catch it. Psyduck evolves into golduck at lvl 33, but you can find Golduck in that water too. For Alakazam, Golem, and Gengar, you have to trade Kadabra, Graveler, and Haunter. Lickitung can be found in the Unknown Dungeon.
For Goldeen, use Good Rod anywhere.
For Seaking, use Super Rod in Unknown Dungeon.
Chansey can be found in the Safari Zone, or routes 12 and 14 or 13 and 14.
Scyther can be found in the Safari Zone or be Purchased at The Game Corner (for a considerable amount of coins though)
Vaporeon, Flareon, Omanyte, and Omastar must be traded.

Weedle, Kakuna, Beedrill- can't be found on this game, can only be traded to get it.
Ekans, Arbok- ?
Raichu- I think if you come across another Pikachu in the power plant, I guess you can evolve it, but I really don't know.
Meowth, Persian- see Weedle, Kakuna, Beedrill.
Psyduck, Golduck- I think if you surf around the water in the Safari Zone, you will find a Psyduck.
Alakazam, Golem, Gengar- You can get them by trading Kadabra, Graveler, and Haunter ,respectively, to another Gameboy, and then get them back from them.
Hitmonchan- Trading is the only way now.
Lickitung- There is some girl in some building on the wooden bridges south of Fuchsia, and she has a Lickitung she is willing to trade for some Pokemon, I don't remember which one though. (Try Tangela, I think that is the one)
Koffing, Weezing- ??????????????????
Goldeen, Seaking- Have you tried a Super Rod?
Chansey, Scyther- Both can be found at the very end area of the Safari Zone, but they are extremely, extremely hard to catch. Chansey, however, can also be found in the Unknown Dungeon, at very high levels I might add. (assuming you have beaten the E4)
Jynx- I don't think so.
Electabuzz, Magmar- No.
Vaporeon, Flareon- see Hitmonchan
Omanyte, Omastar- see Hitmonchan

If you are looking for a way without using Gamesharks or Trading Links, no. You're out of luck. But, you can expand a great deal with this info.

Sorry.

The following Pokemon MUST be traded to get them:
Weedle: (level up to get Kakuna, Beedril)
Ekans: (level up to get Arbok)
Raichu: (or trade other pikachu and give that pikachu thunderstone)
Meowth: (level up to get Persian)
Alakazam: (trade Kadabra it will evolve)
Golem: (trade Graveler it will evolve)
Gengar: (trade Haunter it will evolve)
Hitmonchan: (maybe trade your Hitmonlee for Hitmonchan and then trade back. Then you both have Hitmonchan and Hitmonlee recorded in the Pokedex)
Koffing: (level up to get Weezing)
Vaporeon, Flareon: (could use same trick as Hitmonchan)
Omanyte: (level up to get Omatar)
Electabuzz, Jynx, Magmar, Lickitung

These Pokemon you can get without link cable:

Psyduck: Route 6 (North of Vermilion City), surf in water (level up to get Golduck)
Goldeen: don't know where, mostly caught with Super Rod, some times Good Rod (level it up to get Seaking)
Chansey, Scyther: Safari Zone (rarely seen)

If you knowhow to catch mew, Use the same method to catch the others just fight a different trainer before going back to where Wild mew popped up. Or even Fight A wild pokemon then go back. It will cause a Level 7 Whatever to pop up. The pokemon that pops up depends on the special of the pokemon you faced.