It sounds like somebody in your organization wants to create VLANs w/o understanding the reasons why you'd do it and the pros/cons associated therewith. It sounds like you need to do some measurement and come up with some real reasons for doing this before moving forward, at least with the insane "VLAN for a room" silliness.
You shouldn't start breaking an Ethernet LAN into VLANs unless you have good reasons to do it. The best two reasons are:
Mitigating performance problems. Ethernet LANs can't scale indefinitely. Excessive broadcasts or flooding of frames to unknown destinations will limit their scale. Either of these conditions can be caused by making a single broadcast domain in an Ethernet LAN too big. Broadcast traffic is easy to understand, but flooding of frames to unknown destinations is a bit more obscure (so much so that none of the other posters here even mention it!). If you get so many devices that your switch MAC tables are overflowing switches will be forced to flood non-broadcast frames out all ports if the destination of the frame doesn't match any entries in the MAC table. If you have a large enough single broadcast domain in an Ethernet LAN with a traffic profile that hosts talk infrequently (that is, infrequently enough that their entries have aged out of the MAC tables on your switches) then you can also get excessive flooding of frames.
A desire to limit / control traffic moving between hosts at layer 3 or above. You can do some hackery examining traffic at layer 2 (ala Linux ebtables) but this is difficult to manage (because rules are tied to MAC addresses and changing out NICs necessitates rule changes) can cause what appear to be really, really strange behaviors (doing transparent proxying of HTTP at layer 2, for example, is freaky and fun, but is utterly un-natural and can be very non-intuitive to troubleshoot), and is generally difficult to do at lower layers (because layer 2 tools are like sticks and rocks at dealing with layer 3+ concerns). If you want to control IP (or TCP, or UDP, etc) traffic between hosts, rather than attacking the problem at layer 2, you should subnet and stick firewalls / routers with ACLs between the subnets.
Bandwidth exhaustion problems (unless they're being caused by broadcast packets or flooding of frames) are not solved with VLANs typically. They happen because of a lack of physical connectivity (too few NICs on a server, too few ports in an aggregation group, the need to move up to a faster port speed) and can't be solved by subnetting or deploying VLANs since that won't increase the amount of bandwidth available.
If you don't have even something simple like MRTG running graphing per-port traffic statistics on your switches that's really your first order of business before you start potentially introducing bottlenecks with well-intentioned but uninformed VLAN segmentation. Raw byte counts are a good start, but you should follow it up with targeted sniffing to get more details about the traffic profiles.
Once you know how traffic moves around on your LAN you can begin to think about segmenting the LAN for performance reasons.
If you're really going to try and button down packet and stream-level access between VLANs be prepared to do a lot of legwork with application software and learning / reverse-engineering how it talks over the wire. Limiting access by hosts to servers can often be accomplished with filtering functionality on the servers. Limiting access on the wire can provide a false sense of security and lull administrators into a complacency where they think "Well, I don't need to configure the app. securely because the hosts that can talk to the app. are limited by 'the network'." I'd encourage you to audit the security of your server configuration before I'd start limiting host-to-host communication on the wire.
Typically you create VLANs in Ethernet and map IP subnets 1-to-1 onto them. You're going to need a LOT of IP subnets for what you're describing, and potentially a lot of routing table entries. Better plan those subnets with VLSM to summarize your routing table entries, eh?
(Yes, yes-- there are ways not to use a separate subnet for every VLAN, but sticking in a strictly "plain vanilla" world you'd create a VLAN, think up an IP subnet to use in the VLAN, assign some router an IP address in that VLAN, attach that router to the VLAN, either with a physical interface or a virtual subinterface on the router, connect some hosts to the VLAN and assign them IP addresses in the subnet you defined, and route their traffic in and out of the VLAN.)
Best Answer
If you can "stack" the 2 proposed "core switches", that would be the easiest thing to setup and would allow for "redundancy. You would have the same LAG groups but with ports on each physical switch and then a 2 port LAG to the router at the top.
If you can't stack them, then what you are proposing in your proposal pic is more like VRRP or similar. That takes some pretty decent networking knowledge and the switches have to support such a setup. It's not a hard thing to get setup, but it will take some planning and decent knowledge on how VRRP works.
A "cheap" alternative based on your pics is:
But your question sounds like you could be in over your head a little. If possible, grab a consultant or trusted VAR of your networking equipment (their sales engineers are often happy to help) and work out the best setup for your shop. In the end, you'll learn something and understand it better.